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Notes on Smooth Manifolds and Vector Bundles Aleksey Zinger March 23, 2011 Chapter 0 Notation and Terminology If M is a topological space and p ∈ M , a neighborhood of p in M is an open subset U of M that contains p. The identity element in the groups GLk R and GLk C of invertible k×k real and complex matrices will be denoted Ik . For any set M , idM will denote the identity map on M . If h : M −→ N and f : V −→ X are maps and V ⊂ N , we will denote by f ◦h the map f h h−1 (V ) −→ V −→ X . 1 Chapter 1 Smooth Manifolds and Maps 1 Smooth Manifolds: Definition and Examples Definition 1.1. A topological space M is a topological m-manifold if (TM1) M is Hausdorff and second-countable, and (TM2) every point p ∈ M has a neighborhood U homeomorphic to Rm . A chart around p on M is a pair (U, ϕ), where U is a neighborhood of p in M and ϕ : U −→ U ′ is a homeomorphism onto an open subset of Rm . Thus, the set of rational numbers, Q, in the discrete topology is a 0-dimensional topological manifold. However, the set of real numbers, R, in the discrete topology is not a 0-dimensional manifold because it does not have a countable basis. On the other hand, R with its standard topology is a 1-dimensional topological manifold, since (TM1: R) R is Hausdorff (being a metric space) and second-countable; (TM2: R) the map ϕ = id : U = R −→ R is a homeomorphism; thus, (R, id) is a chart around every point p ∈ R. A topological space satisfying (TM2) in Definition 1.1 is called locally Euclidean; such a space is made up of copies of Rm glued together; see Figure 1.1. While every point in a locally Euclidean space has a neighborhood which is homeomorphic to Rm , the space itself need not be Hausdorff; see Example 1.2 below. A Hausdorff locally Euclidean space is easily seen to be regular, while a regular second-countable space is normal [7, Theorem 32.1], metrizable (Urysohn Metrization Theorem [7, Theorem 34.1]), paracompact [7, Theorem 41.4], and thus admits partitions of unity (see Definition 11.1 below). Example 1.2. Let M = (0×R ⊔ 0′×R)/ ∼, where (0, s) ∼ (0′ , s) for all s ∈ R−0. As sets, M = R ⊔{0′ }. Let B be the collection of all subsets of R ⊔{0′ } of the form (a, b) ⊂ R, a, b ∈ R, (a, b)′ ≡ (a, b) − 0 ⊔ {0′ } if a < 0 < b. This collection B forms a basis for the quotient topology on M . Note that (TO1) any neighborhoods U of 0 and U ′ of 0′ in M intersect, and thus M is not Hausdorff; 2 Rm Rm R− Rm 0′ R+ 0 line with two origins locally Euclidean space Figure 1.1: A locally Euclidean space M , such as an m-manifold, consists of copies of Rm glued together. The line with two origins is a non-Hausdorff locally Euclidean space. (TO2) the subsets M − 0′ and M − 0 of M are open in M and homeomorphic to R; thus, M is locally Euclidean. This example is illustrated in the right diagram in Figure 1.1. The two thin lines have length zero: R− continues through 0 and 0′ to R+ . Since M is not Hausdorff, it cannot be topologically embedded into Rm (and thus cannot be accurately depicted in a diagram). Note that the quotient map q : 0×R ⊔ 0′ ×R −→ M is open (takes open sets to open sets); so open quotient maps do not preserve separation properties. In contrast, the image of a closed quotient map from a normal topological space is still normal [7, Lemma 73.3]. Definition 1.3. A smooth m-manifold is a pair (M, F), where M is a topological m-manifold and F = {(Uα , ϕα )}α∈A is a collection of charts on M such that [ Uα , (SM1) M = α∈A m (SM2) ϕα ◦ϕ−1 β : ϕβ (Uα ∩Uβ ) −→ ϕα (Uα ∩Uβ ) is a smooth map (between open subsets of R ) for all α, β ∈ A; (SM3) F is maximal with respect to (SM2). The collection F is called a smooth structure on M . Since the maps ϕα and ϕβ in Definition 1.3 are homeomorphisms, ϕβ (Uα ∩Uβ ) and ϕα (Uα ∩Uβ ) are open subsets of Rm , and so the notion of a smooth map between them is well-defined; see Figure 1.2. −1 −1 = ϕβ ◦ϕ−1 Since ϕα ◦ϕβ α , smooth map in (SM2) can be replaced by diffeomorphism. If α = β, ϕα ◦ϕ−1 β = id : ϕβ (Uα ∩Uβ ) = ϕα (Uα ) −→ ϕα (Uα ∩Uβ ) = ϕα (Uα ) is of course a smooth map, and so it is sufficient to verify the smoothness requirement of (SM2) only for α 6= β. An element of such a collection F will be called a smooth chart on the smooth manifold on (M, F) or simply M . 3 M Uβ Uα ϕα ϕβ ϕβ (Uβ ) ϕβ (Uα ∩ Uβ ) ϕα ◦ ϕ−1 β ϕα (Uα ∩ Uβ ) ϕα (Uα ) Figure 1.2: The overlap map between two charts is a map between open subsets of Rm . It is hardly ever practical to specify a smooth structure F on a manifold M by listing all elements of F. Instead F can be specified by describing a collection of charts F0 = {(U, ϕ)} satisfying (SM1) and (SM2) in Definition 1.3 and setting F = chart (V, ψ) on M ϕ◦ψ −1 : ψ(U ∩V ) −→ ϕ(U ∩V ) is diffeomorphism ∀ (U, ϕ) ∈ F0 . (1.1) Example 1.4. The map ϕ = id : Rm −→ Rm is a homeomorphism, and thus the pair (Rm , id) is a chart around every point in the topological m-manifold M = Rm . So, the single-element collection F0 = {(Rm , id)} satisfies (SM1) and (SM2) in Definition 1.3. It thus induces a smooth structure F on Rm ; this smooth structure is called the standard smooth structure on Rm . Example 1.5. Every finite-dimensional vector space V has a canonical topology specified by the requirement that any vector-space isomorphism ϕ : V −→ Rm , where m = dim V , is a homeomorphism (with respect to the standard topology on Rm ). If ψ : V −→ Rm is another vector-space isomorphism, then the map ϕ◦ψ −1 : Rm −→ Rm (1.2) is an invertible linear transformation; thus, it is a diffeomorphism and in particular a homeomorphism. So, two different isomorphisms ϕ, ψ : V −→ Rm determine the same topology on V . Each pair (V, ϕ) is then a chart on V , and the one-element collection F0 = {(V, ϕ)} determines a smooth structure F on V . Since the map (1.2) is a diffeomorphism, F is independent of the choice of vector-space isomorphism ϕ : V −→ Rm . Thus, every finite-dimensional vector space carries a canonical smooth structure. Example 1.6. The map ϕ : R −→ R, ϕ(t) = t3 , is a homeomorphism, and thus the pair (R, ϕ) is a chart around every point in the topological 1-manifold M = R. So, the single-element collection F0′ = {(R, ϕ)} satisfies (SM1) and (SM2) in Definition 1.3. It thus induces a smooth structure F ′ on R. While F ′ 6= F, where F is the standard smooth structure on R1 described in Example 1.4, the smooth manifolds (R1 , F) and (R1 , F ′ ) are diffeomorphic in the sense of (2) in Definition 2.1 below. Example 1.7. Let M = S 1 be the unit circle in the complex (s, t)-plane, U+ = S 1 − {i}, U− = S 1 − {−i} . For each p ∈ U± , let ϕ± (p) ∈ R be the s-intercept of the line through the points ±i and p 6= ±i; see Figure 1.3. The maps ϕ± : U± −→ R are homeomorphisms and S 1 = U+ ∪U− . Since U+ ∩ U− = S 1 − {i, −i} = U+ − {−i} = U− − {i} 4 i p ϕ+ (p) ϕ− (p) ϕ+ (s, t) = s 1−t ϕ− (s, t) = s 1+t −1 ∗ ∗ ϕ+ ◦ϕ−1 − : R −→ R , ϕ+ ◦ϕ− (s) = 1/s −i Figure 1.3: A pair of charts on S 1 determining a smooth structure. and ϕ± (U+ ∩U− ) = R−0 ≡ R∗ , the overlap map is ∗ ∗ ϕ+ ◦ϕ−1 − : ϕ− (U+ ∩U− ) = R −→ ϕ+ (U+ ∩U− ) = R ; by a direct computation, this map is s −→ s−1 . Since this map is a diffeomorphism between open subsets of R1 , the collection F0 = (U+ , ϕ+ ), (U− , ϕ− ) determines a smooth structure F on S 1 . A smooth structure on the unit sphere M = S m ⊂ Rm+1 can be defined similarly: take U± ⊂ S m to be the complement of the point q± ∈ S m with the last coordinate ±1 and ϕ± (p) ∈ Rm the intersection of the line through q± and p 6= q± with Rm = Rm × 0. This smooth structure is the unique one with which S m is a submanifold of Rm+1 ; see Definition 5.1 and Corollary 5.8. Example 1.8. Let MB = ([0, 1]×R)/ ∼, (0, t) ∼ (1, −t), be the infinite Mobius Band, U0 = (0, 1)×R ⊂ MB, ϕ1/2 : U1/2 ϕ0 = id : U0 −→ (0, 1)×R, ( (s−1/2, t), = MB−{1/2}×R −→ (0, 1)×R, ϕ1/2 ([s, t]) = (s+1/2, −t), if s ∈ (1/2, 1], if s ∈ [0, 1/2), where [s, t] denotes the equivalence class of (s, t) ∈ [0, 1] × R in MB. The pairs (U0 , ϕ0 ) and (U1/2 , ϕ1/2 ) are then charts on the topological 1-manifold MB. The overlap map between them is ϕ1/2 ◦ϕ−1 0 : ϕ0 (U0 ∩U1/2 ) = (0, 1/2)∪(1/2, 1) ×R −→ ϕ1/2 (U0 ∩U1/2 ) = (0, 1/2)∪(1/2, 1) ×R, ( (s+1/2, −t), if s ∈ (0, 1/2); ϕ1/2 ◦ϕ−1 0 (s, t) = (s−1/2, t), if s ∈ (1/2, 1); see Figure 1.4. Since this map is a diffeomorphism between open subsets of R2 , the collection F0 = (U0 , ϕ0 ), (U1/2 , ϕ1/2 ) determines a smooth structure F on MB. Example 1.9. The real projective space of dimension n, denoted RP n , is the space of real onedimensional subspaces ℓ of Rn+1 (or lines through the origin in Rn+1 ) in the natural quotient topology. In other words, a one-dimensional subspace of Rn+1 is determined by a nonzero vector in 5 0 1 0 1 ϕ1/2 ◦ϕ−1 0 0 1 shift s by − 21 shift s by + 21 , negate t Figure 1.4: The infinite Mobius band MB is obtained from an infinite strip by identifying the two infinite edges in opposite directions, as indicated by the arrows in the first diagram. The two charts on MB of Example 1.8 overlap smoothly. Rn+1 , i.e. an element of Rn+1 −0. Two such vectors determine the same one-dimensional subspace in Rn+1 and the same element of RP n if and only if they differ by a non-zero scalar. Thus, as sets RP n = Rn+1 −0 R∗ ≡ Rn+1 −0 ∼, where c · v = cv ∈ Rn+1 −0 ∀ c ∈ R∗ , v ∈ Rn+1 −0, v ∼ cv ∀ c ∈ R∗ , v ∈ Rn+1 −0. Alternatively, a one-dimensional subspace of Rn+1 is determined by a unit vector in Rn+1 , i.e. an element of S n . Two such vectors determine the same element of RP n if and only if they differ by a non-zero scalar, which in this case must necessarily be ±1. Thus, as sets RP n = S n Z2 ≡ S n ∼, where Z2 = {±1}, Thus, as sets, c · v = cv ∈ S n ∀ c ∈ Z2 , v ∈ S n , v ∼ cv ∀ c ∈ Z2 , v ∈ S 2 . (1.3) RP n = Rn+1 −0 R∗ = S n Z2 . It follows that RP n has two natural quotient topologies; these two topologies are the same, however. The space RP n has a natural smooth structure, induced from that of Rn+1−0 and S n . It is generated by the n+1 charts ϕi : Ui ≡ X0 , X1 , . . . , Xn : Xi 6= 0 −→ Rn , Xi−1 Xi+1 Xn X0 ,..., , ,..., . X0 , X1 , . . . , Xn −→ Xi Xi Xi Xi Note that RP 1 = S 1 . Example 1.10. The complex projective space of dimension n, denoted CP n , is the space of complex one-dimensional subspaces of Cn+1 in the natural quotient topology. Similarly to the real case of Example 1.9, CP n = Cn+1 −0 C∗ = S 2n+1 S 1 , where 1 ∗ 2n+1 n+1 S = c ∈ C : |c| = 1 , S = v∈C −0 : |v| = 1 , c · v = cv ∈ Cn+1 −0 ∀ c ∈ C∗ , v ∈ Cn+1 −0. 6 The two quotient topologies on CP n arising from these quotients are again the same. The space CP n has a natural complex structure, induced from that of Cn+1 −0. There are a number of canonical ways of constructing new smooth manifolds. Proposition 1.11. (1) If (M, F) is a smooth m-manifold, U ⊂ M is open, and F|U ≡ (Uα ∩U, ϕα |Uα ∩U ) : (Uα , ϕα ) ∈ F = (Uα , ϕα ) ∈ F : Uα ⊂ U , then (U, F|U ) is also a smooth m-manifold. (2) If (M, FM ) and (N, FN ) are smooth manifolds, then the collection F0 = (Uα ×Vβ , ϕα ×ψβ ) : (Uα , ϕα ) ∈ FM , (Vβ , ψβ ) ∈ FN (1.4) (1.5) satisfies (SM1) and (SM2) of Definition 1.3 and thus induces a smooth structure on M ×N . It is immediate that the second collection in (1.4) is contained in the first. The first collection is contained in the second because F is maximal with respect to (SM2) in Definition 1.3 and the restriction of a smooth map from an open subset of Rm to a smaller open subset is still smooth. Since every element (Uα , ϕα ) of F is a chart on M , every such element with Uα ⊂ U is also a chart on U . Since {Uα : (Uα , ϕα ) ∈ F} is an open cover of M , {Uα∩U : (Uα , ϕα ) ∈ F} is an open cover of U . Since F satisfies (SM2) in Definition 1.3, so does its subcollection F|U . Since F is maximal with respect to (SM2) in Definition 1.3, so is its subcollection F|U . Thus, F|U is indeed a smooth structure on U . Let m = dim M and n = dim N . Since each (Uα , ϕα ) ∈ FM is a chart on M and each (Vβ , ψβ ) ∈ FN is a chart on N , ϕα ×ψβ : Uα ×Vβ −→ ϕα (Uα )×ψβ (Vβ ) ⊂ Rm ×Rn = Rm+n is a homeomorphism between an open subset of M × N (in the product topology) and an open subset of Rm+n . Since the collections {Uα : (Uα , ϕα ) ∈ FM } and {Vβ : (Vβ , ψβ ) ∈ FN } cover M and N , respectively, the collection Uα ×Vβ : (Uα , ϕα ) ∈ FM , (Vβ , ψβ ) ∈ FN covers M ×N . If (Uα ×Vβ , ϕα ×ψβ ) and (Uα′ ×Vβ ′ , ϕα′ ×ψβ ′ ) are elements of the collection (1.5), Uα ×Vβ ∩ Uα′ ×Vβ ′ = Uα ∩Uα′ × Vβ ∩Vβ ′ , ϕα ×ψβ Uα ×Vβ ∩ Uα′ ×Vβ ′ = ϕα Uα ∩Uα′ × ψβ Vβ ∩Vβ ′ ⊂ Rm+n , ϕα′ ×ψβ ′ Uα ×Vβ ∩ Uα′ ×Vβ ′ = ϕα′ Uα ∩Uα′ × ψβ ′ Vβ ∩Vβ ′ ⊂ Rm+n , and the overlap map, −1 × ϕβ ◦ϕ−1 = ϕα ◦ϕ−1 ϕα ×ψβ ◦ ϕα′ ×ψβ ′ β′ , α′ is the product of the overlap maps for M and N ; thus, it is smooth. So the collection (1.5) satisfies the requirements (SM1) and (SM2) of Definition 1.3 and thus induces a smooth structure on M×N , called the product smooth structure. Corollary 1.12. The general linear group, GLn R = A ∈ Matn×n R : det A 6= 0 , is a smooth manifold of dimension n2 . 7 N M f −1 (V ) f V U ϕ ψ ψ ◦ f ◦ ϕ−1 ϕ(U ) ψ(V ) ϕ(f −1 (V )∩U ) Figure 1.5: A continuous map f between manifolds is smooth if it induces smooth maps between open subsets of Euclidean spaces via the charts. The map 2 det : Matn×n R ≈ Rn −→ R is continuous. Since R−0 is an open subset of R, its pre-image under det, GLn R, is an open subset 2 of Rn and thus is a smooth manifold of dimension n2 by (1) of Proposition 1.11. 2 Smooth Maps: Definition and Examples Definition 2.1. Let (M, FM ) and (N, FN ) be smooth manifolds. (1) A continuous map f : M −→ N is a smooth map between (M, FM ) and (N, FN ) if for all (U, ϕ) ∈ FM and (V, ψ) ∈ FN the map ψ◦f ◦ϕ−1 : ϕ f −1 (V )∩U −→ ψ(V ) (2.1) is a smooth map (between open subsets of Euclidean spaces). (2) A smooth bijective map f : (M, FM ) −→ (N, FN ) is a diffeomorphism if the inverse map, f −1 : (N, FN ) −→ (M, FM ), is also smooth. (3) A smooth map f : (M, FM ) −→ (N, FN ) is a local diffeomorphism if for every p ∈ M there are open neighborhoods Up of p in M and Vp of f (p) in N such that f |Up : Up −→ Vp is a diffeomorphism between the smooth manifolds (Up , FM |Up ) and (Vp , FN |Vp ). If f : M −→ N is a continuous map and (V, ψ) ∈ FN , f −1 (V ) ⊂ M is open and ψ(V ) ⊂ Rn is open, where n = dim N . If in addition (U, ϕ) ∈ FM , then ϕ(f −1 (V )∩U ) is an open subset of Rm , where m = dim M . Thus, (2.1) is a map between open subsets of Rm and Rn , and so the notion of a smooth map between them is well-defined; see Figure 1.5. A bijective local diffeomorphism is a diffeomorphism, and vice versa. In particular, the identity map id : (M, F) −→ (M, F) on any manifold is a diffeomorphism, since for all (U, ϕ), (V, ψ) ∈ FM the map (2.1) is simply ψ◦ϕ−1 : ϕ U ∩V −→ ψ U ∩V ⊂ ψ(V ); 8 it is smooth by (SM2) in Definition 1.3. For the same reason, the map ϕ : U, FM |U −→ ϕ(U ) ⊂ Rm is a diffeomorphism for every (U, ϕ) ∈ FM . A composition of two smooth maps (local diffeomorphisms, diffeomorphisms) is again smooth (a local diffeomorphism, a diffeomorphism). It is generally impractical to verify that the map (2.1) is smooth for all (U, ϕ) ∈ FM and (V, ψ) ∈ FN . The following lemma provides a simpler way of checking whether a map between two smooth manifolds is smooth. Lemma 2.2. Let (M, FM ) and (N, FN ) be smooth manifolds and f : M −→ N a map. (1) If {Uα }α∈A is an open cover of M , then f : M −→ N is a smooth map (local diffeomorphism) if and only if for every α ∈ A the restriction f |Uα : Uα −→ N is a smooth map (local diffeomorphism) with respect to the induced smooth structure on Uα of Proposition 1.11. (2) If FM ;0 and FN ;0 are collections of charts on M and N , respectively, that generate FM and FN in the sense of (1.1), then f : M −→ N is a smooth map (local diffeomorphism) if and only if (2.1) is a smooth map (local diffeomorphism) for every (U, ϕ) ∈ FM ;0 and (V, ψ) ∈ FN ;0 . Thus, f : M −→ N is a smooth map (local diffeomorphism) if and only if (2.1) is a smooth map (local diffeomorphism) for every (U, ϕ) ∈ FM ;0 and all (V, ψ) ∈ FN ;0 in a subcollection of FN ;0 covering f (U ). If follows that for every chart (U, ϕ) ∈ FM the map ϕ : U −→ ϕ(U ) ⊂ Rm is a diffeomorphism. By Lemma 2.2, if f : (M, FM ) −→ (N, FN ) is smooth, then ψ ◦f : f −1 (V ) −→ Rn is also a smooth map from an open subset of M (with the smooth structure induced from FM as in Proposition 1.11) for every (V, ψ) ∈ FN . If in addition f is a diffeomorphism (and thus m = n), ψ◦f ◦ ϕ−1 : ϕ U ∩f −1 (V ) −→ ψ f (U )∩V ⊂ Rm is a diffeomorphism for every (U, ϕ) ∈ FM , and thus (f −1 (V ), ψ◦f ) ∈ FM by the maximality of FM . It follows that every diffeomorphism f : (M, FM ) −→ (N, FN ), which is a map f : M −→ N with certain properties, induces a map f ∗ : FN −→ FM , (V, ψ) −→ f −1 (V ), ψ◦f , which is easily seen to be bijective. However, there are lots of bijections FN −→ FM , and most of them do not arise from a diffeomorphism f : M −→ N (which may not exist at all) or even some map between the underlying spaces. Example 2.3. Let V and W be finite-dimensional vector spaces with the canonical smooth structures of Example 1.5 and f : V −→ W a vector-space homomorphism. If ϕ : V −→ Rm and ψ : W −→ Rn are vector-space isomorphisms, ψ◦f ◦ϕ−1 : Rm −→ Rn is a linear map and thus smooth. Since f (V ) is contained in the domain of ψ, it follows that f : V −→ W is a smooth map. So every homomorphism between finite-dimensional vector spaces is a smooth map with respect to the canonical smooth structures on the vector spaces. 9 Example 2.4. Let Matn×n R be the vector space of n×n real matrices with the canonical smooth structure of Example 1.5. Define f : Matn×n R −→ Matn×n R by A −→ Atr A, (2.2) 2 where Atr is the transpose of A. If ϕ : Matn×n R −→ Rn is an isomorphism of vector spaces (for example, with each component of f sending a matrix to one of its entries), then each component of the map 2 2 ϕ◦f ◦ϕ−1 : Rn −→ Rn 2 is a homogeneous quadratic polynomial on Rn ; so ϕ◦f ◦ϕ−1 is a smooth map. Since the image of f is contained in the domain of ϕ, it follows that the map (2.2) is smooth. The image of f is actually contained in the linear subspace SMatn R of symmetric n×n matrices. Thus, f induces a map f0 : Matn×n R −→ SMatn R, f0 (A) = f (A), obtained by restricting the range of f ; so the diagram SMat nR p77 p f0 p ι p p p f // Matn×n R Matn×n R where ι is the inclusion map, commutes. The induced map f0 is also smooth with respect to the canonical smooth structures on Matn×n R and SMatn R. In fact, if ψ : SMatn R −→ Rn(n+1)/2 is an isomorphism of vector spaces (for example, with each component of f sending a matrix to one of its upper-triangular entries), then each component of the map 2 ψ◦f ◦ϕ−1 : Rn −→ Rn(n+1)/2 2 is again a homogeneous quadratic polynomial on Rn ; so ψ ◦f ◦ϕ−1 is a smooth map and thus f0 is smooth. The smoothness of f0 also follows directly from the smoothness of f because SMatn R is an embedded submanifold of Matn×n R; see Proposition 5.5. Example 2.5. Let (M, FM ) and (N, FN ) be smooth manifolds and FM ×N the product smooth structure on M ×N of Proposition 1.11. Let F0 be as in (1.5). (1) For each q ∈ N , the inclusion as a “horizontal” slice, ιq : M −→ M ×N, p −→ (p, q), is smooth, since for every (U, ϕ) ∈ FM and (U ×V, ϕ×ψ) ∈ F0 with q ∈ V the map ϕ×ψ ◦ιq ◦ϕ−1 = id×ψ(q) : ϕ ι−1 q (U ×V )∩U = ϕ(U ) −→ ϕ×ψ U ×V = ϕ(U )×ψ(V ) is smooth and ιq (U ) ⊂ U ×V . Similarly, for each p ∈ M , the inclusion as a “vertical” slice, ιp : N −→ M ×N, is also smooth. 10 q −→ (p, q), Im ιp N Im ιq q πM πN p M Figure 1.6: A horizontal slice M ×q = Im ιq , a vertical slice p×N = Im ιp , and the two component projection maps M ×N −→ M, N (2) The projection map onto the first component, π1 = πM : M ×N −→ M, (p, q) −→ p, is smooth, since for every (U ×V, ϕ×ψ) ∈ F0 and (U, ϕ) ∈ FM the map −1 −1 (U )∩U ×V = ϕ(U )×ψ(V ) −→ ϕ(U ) = π1 : ϕ×ψ πM ϕ◦πM ◦ ϕ×ψ is smooth (being the restriction of the projection Rm × Rn −→ Rm to an open subset) and πM (U ×V ) ⊂ U . Similarly, the projection map onto the second component, π2 = πN : M ×N −→ N, (p, q) −→ q, is also smooth. The following lemma, corollary, and proposition provide additional ways of constructing smooth structures. Corollary 2.7 follows immediately from Lemmas 2.6 and B.1.1. It gives rise to manifold structures on the tangent and cotangent bundles of a smooth manifold, as indicated in Example 7.5. Lemma 2.6 can be used in the proof of Proposition 2.8. Lemma 2.6. Let M be a Hausdorff second-countable topological space and {ϕα : Uα −→ Mα α∈A a collection of homeomorphisms from open subsets Uα of M to smooth m-manifolds Mα such that (2.3) ϕα ◦ϕ−1 β : ϕβ Uα ∩Uβ −→ ϕα Uα ∩Uβ is a smooth map for all α, β ∈ A. If the collection {Uα }α∈A covers M , then M admits a unique smooth structure such that each map ϕα is a diffeomorphism. Corollary 2.7. Let M be a set and {ϕα : Uα −→ Mα α∈A a collection of bijections from subsets Uα of M to smooth m-manifolds Mα such that ϕα ◦ϕ−1 β : ϕβ Uα ∩Uβ −→ ϕα Uα ∩Uβ is a smooth map between open subsets of Mβ and Mα , respectively, for all α, β ∈ A. If the collection {Uα }α∈A separates points in M and a countable subcollection {Uα }α∈A0 of {Uα }α∈A covers M , then M admits a unique topology TM and smooth structure FM such that each map ϕα is a diffeomorphism. 11 Proposition 2.8. If a group G acts properly discontinuously on a smooth m-manifold (M̃ , FM̃ ) by diffeomorphisms and π : M̃ −→ M = M̃ /G is the quotient projection map, then F0 = (π(U ), ϕ◦{π|U }−1 ) : (U, ϕ) ∈ FM̃ , π|U is injective is a collection of charts on the quotient topological space M that satisfies (SM1) and (SM2) in Definition 1.3 and thus induces a smooth structure FM on M . This smooth structure on M is the unique one satisfying either of the following two properties: (QSM1) the projection map M̃ −→ M is a local diffeomorphism; (QSM2) if N is a smooth manifold, a continuous map f : M −→ N is smooth if and only if the map f ◦π : M̃ −→ N is smooth. In the case of Lemma 2.6, ϕα (Uα∩Uβ ) is an open subset of Mα because Uα and Uβ are open subsets of M and ϕα is a homeomorphism; thus, smoothness for the map (2.3) is a well-defined requirement in light of (1) of Proposition 1.11 and (1) of Definition 2.1. In the case of Corollary 2.7, ϕα (Uα ∩ Uβ ) need not be a priori open in Mα , and so this must be one of the assumptions. In both cases, the requirement that ϕα ◦ϕ−1 β be smooth can be replaced by the requirement that it be a diffeomorphism. We leave proofs of Lemma 2.6, Corollary 2.7, and Proposition 2.8 as exercises. The smooth structure FM on M of Proposition 2.8 is called the quotient smooth structure on M . For example, the group Z acts on R and on R×R by Z × R −→ R, Z × R×R −→ R×R, (m, s) −→ s + m, m (m, s, t) −→ s+m, (−1) t . (2.4) (2.5) Both of these actions satisfy the assumptions of Proposition 2.8 and thus give rise to quotient smooth structures on S 1 = R/Z and MB = (R × R)/Z. These smooth structures are the same as those of Examples 1.7 and 1.8, respectively. Example 1.6 is a special case of the following phenomenon. If (M, F) is a smooth manifold and h : M −→ M is a homeomorphism, then h∗ F ≡ h−1 (U ), ϕ◦h) : (U, ϕ) ∈ F is also a smooth structure on M , since the overlap maps are the same as for the collection F. The smooth structures F and h∗ F are the same if and only if h : (M, F) −→ (M, F) is a diffeomorphism. However, in all cases, the map h−1 : (M, F) −→ (M, h∗ F) is a diffeomorphism; so if a topological manifold admits a smooth structure, it admits many smooth equivalent (diffeomorphic) smooth structures. This raises the question of which topological manifolds admit smooth structures and if so how many inequivalent ones. Since every connected component of a topological manifold is again a topological manifold, it is sufficient to study this question for connected topological manifolds. dim=0: every connected topological 0-manifold M consists of a single point, M = {pt}; the only smooth structure on such a topological manifold is the single-element collection {(M, ϕ)}, where ϕ is the unique map M −→ R0 . 12 dim=1: every connected topological (smooth) 1-manifold is homeomorphic (diffeomorphic) to either R or S 1 in the standard topology (and with standard smooth structure); a short proof of the smooth statement is given in [4, Appendix]. dim=2: every topological 2-manifold admits a unique smooth structure; every compact topological 2-manifold is homeomorphic (and thus diffeomorphic) to either a “torus” with g ≥ 0 handles or to a connected sum of such a “torus” with RP 2 [7, Chapter 8]; every such manifold admits a smooth structure as it is the quotient of either S 2 or R2 by a group acting properly discontinuously by diffeomorphisms. dim=3: every topological 3-manifold admits a unique smooth structure [5]. dim=4: there are lots of topological 4-manifolds that admit no smooth structure and lots of other topological 4-manifolds (including R4 ) that admit many (even uncountably many) smooth structures. The first known example of a topological manifold admitting non-equivalent smooth structures is the 7-sphere [3]. Since then the situation in dimensions 5 or greater has been sorted out by topological arguments [8]. Remark 2.9. While topology studies the topological category T C, differential geometry studies the smooth category SC. The objects in the latter are smooth manifolds, while the morphisms are smooth maps. The composition of two morphisms is the usual composition of maps (which is still a smooth map). For each object (M, FM ), the identity morphism is just the identity map idM on M (which is a smooth map). The “forgetful map” SC −→ T C, (M, FM ) −→ M, f : (M, FM ) −→ (N, FN ) −→ f : M −→ N , is a functor from the smooth category to the topological category. In the remainder of these notes, we will typically denote a smooth manifold in the same way as its underlying set and topological space; so a smooth manifold M will be understood to come with a smooth structure FM . 3 Tangent Vectors If M is an m-manifold embedded in Rn , with m ≤ n, and γ : (a, b) −→ M is a smooth map (curve on M ), then γ(t+τ ) − γ(t) ∈ Rn (3.1) γ̇(t) ≡ lim τ −→0 τ should be a tangent vector of M at γ(t). The set of such vectors is an m-dimensional linear subspace of Rn ; it is often thought of as having the 0-vector at p; see Figure 1.7. However, this presentation of the tangent space Tp M of M at p depends on the embedding of M in Rn , and not just on M and p. On the other hand, the tangent space at a point p ∈ Rm should be Rm itself, but based (with the origin) at p. Each vector v ∈ Rm acts on smooth functions f defined near p by ∂v |p f = lim t−→0 f (p+tv) − f (p) . t 13 (3.2) γ̇(t) p Tp S 1 Figure 1.7: The tangent space of S 1 at p viewed as a subspace of R2 . If v = ei is the i-th coordinate vector on Rm , then ∂v |p f is just the i-th partial derivative ∂i f |p of f at p. The map ∂v |p defined by (3.2) takes each smooth function defined on a neighborhood of p in Rm to R and satisfies: (TV1) if f : U −→ R and g : V −→ R are smooth functions on neighborhoods of p such that f |W = g|W for some neighborhood W of p in U ∩V , then ∂v |pf = ∂v |p g; (TV2) if f : U −→ R and g : V −→ R are smooth functions on neighborhoods of p and a, b ∈ R, then ∂v |p af +bg = a ∂v |p f + b ∂v |p g , where af +bg is the smooth function on the neighborhood U ∩V given by {af +bg}(q) = af (q) + bg(q) ; (TV3) if f : U −→ R and g : V −→ R are smooth functions on neighborhoods of p, then ∂v |p (f g) = f (p)∂v |p g + g(p)∂v |p f , where f g is the smooth function on the neighborhood U ∩V given by {f g}(q) = f (q)g(q). It turns out every that R-valued map on the space of smooth functions defined on neighborhood of p satisfying (TV1)-(TV3) is ∂v |p for some v ∈ Rm ; see Proposition 3.4 below. At the same time, these three conditions make sense for any smooth manifold, and this approach indeed leads to an intrinsic definition of tangent vectors for smooth manifolds. The space of functions defined on various neighborhoods of a point does not have a very nice structure. In order to study the space of operators satisfying (TV1)-(TV3) it is convenient to put an equivalence relation on this space. Definition 3.1. Let M be a smooth manifold and p ∈ M . (1) Functions f : U −→ R and g : V −→ R defined on neighborhoods of p in M are p-equivalent, or f ∼p g, if there exists a neighborhood W of p in U ∩V such that f |W = g|W . (2) The set of p-equivalence classes of smooth functions is denoted F̃p ; the p-equivalence class of a smooth function f : U −→ R on a neighborhood of p is called the germ of f at p and is denoted f p . 14 The set F̃p has a natural R-algebra structure: af p + bg p = af +bgp , f p · gp = f gp ∀ f p , g p ∈ F̃p , a, b ∈ R, where af+bg and f g are functions defined on U∩V if f and g are defined on U and V , respectively. There is a well-defined valuation homomorphism, f p −→ f (p). evp : F̃p −→ R, Let Fp = ker evp ; this subset of F̃p consists of the germs at p of the smooth functions defined on neighborhoods of p in M that vanish at p. Since evp is an R-algebra homomorphisms, Fp is an ideal in F̃p ; this can also be seen directly: if f (p) = 0, then {f g}(p) = 0. Let Fp2 ⊂ Fp be the ideal in F̃p consisting of all finite linear combinations of elements of the form f p g p with f p , g p ∈ Fp . If c ∈ R, let cp ∈ F̃p denote the germ at p of the constant function with value c on M . Lemma 3.2. Let M be a smooth manifold and p ∈ M . If v is a derivation on F̃p relative to the valuation evp ,1 then v|Fp2 ≡ 0, v(cp ) = 0 ∀ c ∈ R. (3.3) If f p , g p ∈ Fp , then f (p), g(p) = 0 and thus v f p g p = f (p)v(g p ) + g(p)v(f p ) = 0; so v vanishes identically on Fp2 . If c ∈ R, v(cp ) = v(1p cp ) = 1(p) · v(cp ) + c(p) · v(1p ) = 1 · v(cp ) + c · v(1p ) = v(cp ) + v(c · 1p ) = v(cp ) + v(cp ); so v(cp ) = 0. Corollary 3.3. If M is a smooth manifold and p ∈ M , the map v −→ v|Fp induces an isomorphism from the vector space Der(F̃p , evp ) of derivations on F̃p relative to the valuation evp to ∗ L ∈ Hom(Fp , R) : L|Fp2 ≡ 0 ≈ Fp /Fp2 . The set Der(F̃p , evp ) of derivations on F̃p relative to the valuation evp indeed forms a vector space: av + bw f p = av f p + bw f p ∀ v, w ∈ Der(F̃p , evp ), a, b ∈ R, f p ∈ F̃p . If v ∈ Der(F̃p , evp ), the restriction of v to Fp ⊂ F̃p is a homomorphism to R that vanishes on Fp2 by Lemma 3.2. Conversely, if L : Fp −→ R is a linear homomorphism vanishing on Fp2 , define vL : F̃p −→ R 1 i.e. v : F̃p −→ R is an R-linear map such that by vL f p = L f −f (p)p ; v(f p g p ) = evp (f p )v(g p ) + evp (g p )v(f p ) 15 ∀ f p , g p ∈ F̃p . since the function f −f (p) vanishes at p, f −f (p)p ∈ Fp and so vL is well-defined. It is immediate that vL is a homomorphism of vector spaces. Furthermore, for all f p , g p , vL f p g p = L f g−f (p)g(p)p = L f (p)g−g(p)p + g(p)f −f (p)p + f −f (p)p g−g(p)p = f (p)L g−g(p)p + g(p)L f −f (p)p + L f −f (p)p g−g(p)p = f (p)vL g p + g(p)vL f p + 0, since L vanishes on Fp2 ; so vL is a derivation with respect to the valuation evp . It is also immediate that the maps Der(F̃p , evp ) −→ L ∈ Hom(Fp , R) : L|Fp2 ≡ 0 , v −→ Lv ≡ v|Fp , (3.4) L ∈ Hom(Fp , R) : L|Fp2 ≡ 0 −→ Der(F̃p , evp ), L −→ vL , are homomorphisms of vector spaces. If L ∈ Hom(Fp , R) and L|Fp2 ≡ 0, the restriction of vL to Fp is L, and so LvL = L. If v ∈ Der(F̃p , evp ) and f p ∈ F̃p , by the second statement in (3.3) v f p = v f p − v f (p)p = v f −f (p)p = Lv f −f (p)p = vLv f p ; so vLv = v and the two homomorphisms in (3.4) are inverses of each other. This completes the proof of Corollary 3.3. Proposition 3.4. If p ∈ Rm , the vector space Fp /Fp2 is m-dimensional and the homomorphism ∗ Rm −→ Der F̃p , evp ≈ Fp /Fp2 , v −→ ∂v |p , (3.5) induced by (3.2), is an isomorphism. By (TV1), ∂v |p induces a well-defined map F̃p −→ R. By (TV2), ∂v |p is a vector-space homomorphism. By (TV3), ∂v |p is a derivation with respect to the valuation evp . Thus, the map (3.5) is well-defined and is clearly a vector-space homomorphism. If πj : Rm −→ R is the projection to the j-th component, ( 1, if i = j; (3.6) ∂ei |p πj −πj (p) = ∂i (πj −πj (p)) p = δij ≡ 0, if i 6= j. Thus, the homomorphism (3.5) is injective, and the set {πj −πj (p) } is linearly independent in Fp /Fp2 . On the other hand, Lemma 3.5 below implies that f (p+x) = f (p) + i=m X (∂i f )p xi + i=1 i,j=m X i,j=1 Z p 1 xi xi (1−t)(∂i ∂j f )p+tx dt (3.7) 0 for every smooth function f defined on a open ball U around p in Rm and for all p+x ∈ U . Thus, the set {πj −πj (p) } spans Fp /Fp2 ; so Fp /Fp2 is m-dimensional and the homomorphism (3.5) is an p isomorphism. Note that the inverse of the isomorphism (3.5) is given by v −→ v(π1 p ), . . . , v(πm p ) ; Der F̃p , evp −→ Rm , by (3.6), this is a right inverse and thus must be the inverse. 16 (3.8) Lemma 3.5. If h : U −→ R is a smooth function defined on an open ball U around a point p in Rm , then i=m X Z 1 xi (∂i h)p+tx dt h(p+x) = h(p) + i=1 0 for all p+x ∈ U . This follows from the Fundamental Theorem of Calculus: h(p+x) = h(p) + Z 0 1 d h(p+tx)dt = h(p) + dt = h(p) + Z 1 i=m X (∂i h)p+tx xi dt 0 i=1 i=m X Z 1 (∂i h)p+tx dt. xi i=1 0 Corollary 3.6. If M is a smooth m-manifold and p ∈ M , the vector space Der(F̃p , evp ) is mdimensional. If ϕ : U −→ Rm is a smooth chart around p ∈ M , the map f −→ f ◦ ϕ induces an R-algebra isomorphism (3.9) ϕ∗ : F̃ϕ(p) −→ F̃p , f ϕ(p) −→ f ◦ϕp . Since evϕ(p) = evp ◦ϕ∗ , ϕ∗ restricts to an isomorphism Fϕ(p) −→ Fp and descends to an isomorphism 2 Fϕ(p) /Fϕ(p) −→ Fp /Fp2 . (3.10) Thus, Corollary 3.6 follows from Corollary 3.3 and Proposition 3.4. Definition 3.7. Let M be a smooth manifold and p ∈ M . (1) The tangent space of M at p is the vector space Tp M = Der(F̃p , evp ); a tangent vector of M at p is an element of Tp M . (2) The cotangent space of M at p is Tp∗ M ≡ (Tp M )∗ ≡ Hom(Tp M, R). By Corollary 3.6, Tp M and Tp∗ M are m-dimensional vector spaces if M is an m-dimensional manifold. By Proposition 3.4, Tp Rm is canonically isomorphic to Rm for every p ∈ Rm . By Corollary 3.3, Tp∗ M ≈ Fp /Fp2 ; an element f p +Fp2 of Fp /Fp2 determines the vector-space homomorphism v −→ v f p . Tp M −→ R, (3.11) Any smooth function f defined on a neighborhood of p in M defines an element of T ∗ M in the same way, but this element depends only on f −f (p)p +Fp2 ∈ Fp /Fp2 . Example 3.8. Let V be an m-dimensional vector space with the canonical smooth structure of Example 1.5. For p, v ∈ V , let ∂v |p : F̃p −→ R 17 be the derivation with respect to evp defined by (3.2). The homomorphism V −→ Tp V = Der(F̃p , evp ), v −→ ∂v |p , (3.12) is injective because for any linear functional f : V −→ R ∂v |p f = lim t−→0 f (p+tv) − f (p) f (p)+tf (v) − f (p) = lim = f (v); t−→0 t t so ∂v |p f 6= 0 on every linear functional f on V not vanishing on v and thus ∂v |p 6= 0 ∈ Tp V if v 6= 0 (such functionals exist if v 6= 0; they are smooth by Example 2.3). Since the dimension of Tp V is m by Corollary 3.6, the homomorphism (3.12) is an isomorphism of vector spaces. So, for every finite-dimensional vector space V and p ∈ Tp V , (3.12) provides a canonical identification of Tp V with V (but not with Rm ); the dual of (3.12) provides a canonical identification Tp∗ V with V ∗ = Hom(V, R). 4 Differentials of Smooth Maps Definition 4.1. Let h : M −→ N be a smooth map between smooth manifolds and p ∈ M . (1) The differential of h at p is the map dp h : Tp M −→ Th(p) N, dp h(v) f h(p) = v f ◦hp (2) The pull-back map on the cotangent spaces is the map ∗ ∗ h∗ ≡ dp h : Th(p) N −→ Tp∗ M, ∀ v ∈ Tp M, f h(p) ∈ F̃h(p) . η −→ η ◦ dp h . (4.1) (4.2) The map (4.1) is a vector-space homomorphism, and thus so is h∗ . It is immediate from the definition that dp idM = idTp M and thus id∗M = idTp∗ M . If N = R, Th(p) R is canonically isomorphic to R, via the map Th(p) R −→ R, w −→ w idR ; see (3.8). In particular, if v ∈ Tp M , dp h(v) −→ dp h(v) idR ≡ v idR ◦h = v(h). Thus, under the canonical identification of Th(p) R with R, the differential dp h of a smooth map h : M −→ R is given by dp h(v) = v(h) ∀ v ∈ Tp M (4.3) and so corresponds to the same element of Tp∗ M as h−h(p)p + Fp2 ∈ Fp /Fp2 ; see (3.11). 18 Example 4.2. Let V and W be finite-dimensional vector spaces with the canonical smooth structures of Example 1.5. By Example 2.3, every vector-space homomorphism h : V −→ W is smooth. If p, v ∈ V and f : U −→ R is a smooth function defined on a neighborhood of h(p) in W , by (4.1) and (3.2) dp h ∂v |p f (h(p+tv)) − f (h(p)) t−→0 t f (h(p)+th(v)) − f (h(p)) = ∂h(v) |h(p) (f ) . = lim t−→0 t (f ) = ∂v |p (f ◦h) = lim Thus, under the canonical identifications of Tp V with V and Th(p) W with W as in Example 3.8, the differential dp h at p of a vector-space homomorphism h : V −→ W corresponds to the homomorphism h itself; so the diagram Tp V dp h // Th(p) W (3.12) (4.4) (3.12) V h // W commutes. In particular, the differentials of an identification ϕ : V −→ Rm induce the same identifications on the tangent spaces. Lemma 4.3. If g : M −→ N and h : N −→ X are smooth maps between smooth manifolds and p ∈ M , then dp (h◦g) = dg(p) h ◦ dp g : Tp M −→ Th(g(p)) X. (4.5) ∗ Thus, (h◦g)∗ = g ∗ ◦h∗ : Th(g(p)) X −→ Tp∗ M and g∗ dg(p) f = dp (f ◦g) (4.6) whenever f is a smooth function on a neighborhood of g(p) in N . If v ∈ Tp M and f is a smooth function on a neighborhood of h(g(p)) in X, then by (4.1) {dp (h◦g)}(v) (f ) = v f ◦h◦g = dp g(v) (f ◦h) = dg(p) h dp g(v) (f ) = {dg(p) h◦dp g}(v) (f ); thus, (4.5) holds. The second claim is the dual of the first. For the last claim, note that g∗ dg(p) f ≡ dg(p) f ◦ dp g = dp (f ◦g) (4.7) by (4.2) and (4.5). For the purposes of applying (4.2) and (4.5), all expressions in (4.7) are viewed as maps to Tf (g(p)) R, before its canonical identification with R. The identities of course continue to hold after this identification. Definition 4.4. A smooth curve in a smooth manifold M is a smooth map γ : (a, b) −→ M , where (a, b) is a nonempty open (possibly infinite) interval in R. The tangent vector to a smooth curve γ : (a, b) −→ M at t ∈ (a, b) is the vector γ ′ (t) = d γ(t) ≡ dt γ ∂e1 |t ∈ Tγ(t) M, dt where e1 = 1 ∈ R1 is the oriented unit vector. 19 (4.8) If h : M −→ N is a smooth map between smooth manifolds and γ : (a, b) −→ M is a smooth curve in M , then h◦γ : (a, b) −→ N is a smooth curve on N and by the chain rule (4.5) (h◦γ)′ (t) ≡ dt {h◦γ} ∂e1 |t = dγ(t) h ◦ dt γ ∂e1 |t = dγ(t) h dt γ(∂e1 |t ) = dγ(t) γ ′ (t) ∈ Th(γ(t)) N (4.9) for every t ∈ (a, b). Lemma 4.5. Let V be a finite-dimensional vector space with its canonical smooth structure of Example 1.5. If γ : (a, b) −→ V is a smooth curve and t ∈ (a, b), γ ′ (t) ∈ Tγ(t) V corresponds to γ̇(t) = lim τ −→0 γ(t+τ ) − γ(t) ∈V τ under the canonical isomorphism Tγ(t) V ≈ V provided by (3.12). If h : V −→ W is a homomorphism of vector spaces, ḋ h◦γ(t) = h(γ̇(t)) (4.10) by the linearity of h. Thus, by (4.9) and the commutativity of (4.4), it is sufficient to prove this lemma for V = Rm , which we now assume to be the case. If f : U −→ R is a smooth function defined on a neighborhood of γ(t) in Rm , by (4.8), (4.1), the usual multi-variable chain rule, and (3.1) f (γ(t+τ )) − f (γ(t)) γ ′ (t)}(f ) = dt γ ∂e1 |t (f ) = ∂e1 |t (f ◦γ) = lim τ −→0 τ f (γ(t)+τ γ̇(t)) − f (γ(t)) = J (f )γ(t) γ̇(t) = lim τ −→0 τ = ∂γ̇(t) |γ(t) f , (4.11) where J (f )γ(t) : Rm −→ R is the usual Jacobian (matrix of first partials) of the smooth map f from an open subset of Rm to R evaluated at γ(t). Thus, under the canonical identification of Tγ(t) Rm with Rm provided by (3.5), the tangent vector γ ′ (t) of Definition 4.4 corresponds to the tangent vector γ̇(t) of calculus. Corollary 4.6. Let (M, FM ) be a smooth manifold. For every p ∈ M and v ∈ Tp M , there exists a smooth curve γ : (a, b) −→ M s.t. γ(0) = p, γ ′ (0) = v. If ϕ : U −→ Rm is a smooth chart around p on M , the homomorphism dϕ(p) ϕ−1 : Tϕ(p) Rm −→ Tp M is an isomorphism. Thus, by (4.9), it is sufficient to prove the claim for V = Rm , which we now assume to be the case. By Lemma 4.5 applied with V = Rm , it is to show that for all p, v ∈ Rm there exists a smooth curve γ : (a, b) −→ Rm s.t. γ(0) = p, γ̇(0) = v. An example of such a curve is given by γ : (−∞, ∞) −→ Rm , 20 t −→ p+tv. Example 4.7. By Example 2.4, the map h : Matn×n R −→ SMatn R, h(A) = Atr A, is smooth; we determine the homomorphism dIn h. The map γ : (−∞, ∞) −→ Matn×n R, t −→ In +tA, is a smooth curve such that γ(0) = In and γ̇(0) = A. By (4.10), the homomorphism induced by dIn h via the isomorphisms provided by (3.12) takes γ̇(0) = A to z}|{ ˙ h(γ(t)) − h(γ(0)) (In +tA)tr (In +tA) − In h◦γ(0) ≡ lim = lim = A + Atr . t−→0 t−→0 t t Thus, the homomorphism induced by dIn h via the identifications provided by (3.12) is given by dIn h : Matn×n R −→ SMatn R, A −→ A + Atr . In particular, dIn h is surjective, because its restriction to the subspace SMatn R ⊂ Matn×n R is. Let ϕ = (x1 , . . . , xm ) : U −→ Rm be a smooth chart on a neighborhood of a point p in M ; so, xi = πi ◦ ϕ, where πi : Rm −→ R is the projection to the i-th component as before. Since the 2 , map (3.9) induces the isomorphism (3.10) and {πi −xi (p)ϕ(p) }i is a basis for Fϕ(p) /Fϕ(p) ϕ∗ {πi −xi (p)ϕ(p) }i ≡ (πi −xi (p))◦ϕp i = xi −xi (p)p i is a basis for Fp /Fp2 . Thus, {dp xi }i is a basis for Tp∗ M , since dp xi and xi −xi (p)p act in the same way on all elements of Tp M ; see the paragraph following Definition 4.1. For each i = 1, 2, . . . , m, let ∂ = dϕ(p) ϕ−1 ∂ei |ϕ(p) ∈ Tp M. (4.12) ∂xi p By (4.1), for every smooth function f defined on a neighborhood of p in M ∂ −1 −1 (f ) = d ϕ ∂ | (f ) = ∂ | f ◦ϕ e e ϕ(p) ϕ(p) ϕ(p) i i ∂xi p −1 = ∂i f ◦ϕ (4.13) )|ϕ(p) is the i-th partial derivative of the function f ◦ϕ−1 at ϕ(p); this is a smooth function defined on a neighborhood of ϕ(p) in Rm . In particular, for all i, j = 1, 2, . . . , m ∂ ∂ = (xj ) = ∂i πj ◦ϕ ◦ ϕ−1 = δij ; dp xj ∂xi p ∂xi p ∂ the first equality above is a special case of (4.3). Thus, { ∂x }i is a basis for Tp M ; it is dual to the i p basis {dp xi }i for Tp∗ M . The coefficients of other elements of Tp M and Tp∗ M with respect to these bases are given by i=m i=m X X ∂ ∂ = dp xi (v) v= v(xi ) ∀ v ∈ Tp M ; (4.14) ∂xi p ∂xi p i=1 i=1 i=m X ∂ d p xi ∀ η ∈ Tp∗ M. (4.15) η η= ∂xi p i=1 21 The first identities in (4.14) and (4.15) are immediate from the two bases being dual to each other (each dp xj gives the same values when evaluated on both sides of the first identity in (4.14); both sides of (4.15) evaluate to the same number on each ∂x∂ j p ). The second equality in (4.14) follows from (4.3). If f is a smooth function on a neighborhood of p, by (4.15), (4.3), and (4.13) i=m X dp f = i=1 i=m i=m X ∂ X ∂ ∂i (f ◦ϕ−1 ) ϕ(p) dp xi . d x = (f ) d x = dp f p i p i ∂xi p ∂xi p i=1 (4.16) i=1 If ψ = (y1 , . . . , ym ) : V −→ Rm is another smooth chart around p, by (4.14), (4.3), and (4.13) ⇐⇒ i=m i=m X ∂ X ∂ ∂ −1 (yi ) ∂ = = ∂ (π ◦ψ◦ϕ ) j i ϕ(p) ∂xj p ∂xj p ∂yi p ∂yi p i=1 i=1 ∂ ∂ ∂ ∂ ,..., = ,..., J ψ◦ϕ−1 ϕ(p) , ∂x1 p ∂xn p ∂y1 p ∂yn p (4.17) where J (ψ◦ϕ−1 )ϕ(p) is the Jacobian of the smooth map ψ◦ϕ−1 between the open subsets ϕ(U ∩V ) and ψ(U ∩V ) of Rm at ϕ(p); see Figure 1.2 with ϕα = ψ and ϕβ = ϕ. Suppose next that f : M −→ N is a map between smooth manifolds and ϕ = (x1 , . . . , xm ) : U −→ Rm and ψ = (y1 , . . . , yn ) : V −→ Rn are smooth charts around p ∈ M and f (p) ∈ N , respectively; see Figure 1.5. By (4.14), (4.1), and (4.13), X i=n i=n X ∂ ∂ ∂ ∂ ∂ yi ◦f ) = (yi ) = dp f dp f ∂xj p ∂xj p ∂yi f (p) ∂xj p ∂yi f (p) i=1 i=1 i=n X ∂ −1 ∂j (πi ◦ψ◦f ◦ϕ ) ϕ(p) = ; ∂yi f (p) (4.18) i=1 so the matrix of the linear transformation dp f : Tp M −→ Tf (p) N with respect to the bases { ∂x∂ j p }j and { ∂ }i is J (ψ◦f ◦ϕ−1 )ϕ(p) , the Jacobian of the smooth map ψ◦f ◦ϕ−1 between the open ∂yi f (p) subsets ϕ(U ∩ f −1 (V )) and ψ(V ) of Rm and Rn , respectively, evaluated at ϕ(p). In particular, dp f is injective (surjective) if and only if J (ψ ◦ f ◦ ϕ−1 )ϕ(p) is. The f = id case of (4.18) is the change-of-coordinates formula (4.17). If M and N are open subsets of Rm and Rn , respectively, ϕ = idM , and ψ = idN , then under the canonical identifications Tp Rm = Rm and Tf (p) Rn = Rn the differential dp f is simply the Jacobian J (f )p of f at p. The chain-rule formula (4.5) states that the Jacobian of a composition of maps is the (matrix) product of the Jacobians of the maps; if M , N , and X are open subsets of Euclidean spaces, this yields the usual chain rule for smooth maps between open subsets of Euclidean spaces, for free (once it is checked that all definitions above make sense and correspond to the standard ones for Euclidean spaces). By the above, if ϕ = (x1 , . . . , xm ) : U −→ Rm is a smooth chart around a point p in M , then {dp xi }i is a basis for Tp∗ M . A weak converse to this statement is true as well; see Corollary 4.10 below. The key tool in obtaining it is the Inverse Function Theorem for Rm ; see [6, Theorem 8.3], for example. 22 Theorem 4.8 (Inverse Function Theorem). Let U ′ ⊂ Rm be an open subset and f : U ′ −→ Rm a smooth map. If the Jacobian J (f )p of f is non-singular for some p ∈ U ′ , there exist neighborhoods U of p in U ′ and V of f (p) in Rm such that f : U −→ V is a diffeomorphism. Corollary 4.9 (Inverse Function Theorem for Manifolds). Let f : M −→ N be a smooth map between smooth manifolds. If the differential dp f : Tp M −→ Tf (p) N is an isomorphism for some p ∈ M , then there exist neighborhoods U of p in M and V of f (p) in N such that f : U −→ V is a diffeomorphism. Let ϕ = (x1 , . . . , xm ) : U ′ −→ Rm and ψ = (y1 , . . . , ym ) : V ′ −→ Rm be smooth charts around p in M and f (p) in N , respectively; see Figure 1.5. Then, ψ◦f ◦ϕ−1 : ϕ U ′ ∩f −1 (V ′ ) −→ ϕ(V ′ ) ⊂ Rm is a smooth map from an open subset of Rm to Rm such that J (ψ◦f◦ϕ−1 )ϕ(p) is non-singular (since by (4.18) this is the matrix of the linear transformation dp f with respect to bases { ∂x∂j |p }j and { ∂y∂ i |f (p) }i ). Since ϕ and ψ are homeomorphisms onto the open subsets ϕ(U ′ ) and ψ(V ′ ) of Rm , by Theorem 4.8 there exist open neighborhoods U of p in U ′ ∩f −1 (V ′ ) and V of f (p) in V ′ such that the restriction ψ◦f ◦ϕ−1 : ϕ(U ) −→ ψ(V ) is a diffeomorphism. Since ϕ : U −→ ϕ(U ) and ψ : V −→ ψ(V ) are also diffeomorphisms, it follows that so is f : U −→ V (being composition of ψ◦f ◦ϕ−1 with ψ −1 and ϕ). Corollary 4.10. Let M be a smooth m-manifold. If x1 , . . . , xm : U ′ −→ R are smooth functions such that {dp xi }i is a basis for Tp∗ M for some p ∈ U ′ , then there exists a neighborhood U of p in U ′ such that ϕ = (x1 , . . . , xm ) : U −→ R is a smooth chart around p. Let f = (x1 , . . . , xm ) : U ′ −→ Rm . Since {dp xi }i is a basis for Tp∗ M , the differential d p x1 dp f = ... : Tp M −→ Rm dp xm is an isomorphism (for each v ∈ Tp M −0, there exists i such that dp xi (v) 6= 0). Thus, Corollary 4.10 follows immediately from Corollary 4.9 with M = U ′ and N = Rm . Corollary 4.11. Let M be a smooth m-manifold. If x1 , . . . , xn : U ′ −→ R are smooth functions such that the set {dp xi }i spans Tp∗ M for some p ∈ U ′ , then there exists a neighborhood U of p in U ′ such that an m-element subset of {xi }i determines a smooth chart around p on M . This claim follows from Corollary 4.10 by choosing a subset of {xi }i so that the corresponding subset of {dp xi }i is a basis for Tp∗ M . Corollary 4.12. Let M be a smooth m-manifold. If x1 , . . . , xk : U ′ −→ R are smooth functions such that the set {dp xi }i is linearly independent in Tp∗ M for some p ∈ U ′ , then there exist a neighborhood U of p in U ′ and a set of smooth functions xk+1 , . . . , xm : U −→ R such that the map ϕ = x1 , . . . , xk , xk+1 , . . . , xn ) : U −→ Rm is a smooth chart around p on M . 23 This claim follows from Corollary 4.10 by choosing a smooth chart ψ = (y1 , . . . , ym ) : U ′′ −→ Rm on a neighborhood U ′′ of p in U ′ and adding some of the functions yj to the set {xi }i so that the corresponding set {dp xi , dp yj } is a basis for Tp∗ M . Remark 4.13. The differential of a smooth map induces a functor from the category of pointed smooth manifolds (smooth manifolds with a choice of a point) and pointed smooth maps (smooth maps taking chosen points to each other) to the category of finite-dimensional vector spaces and vector-space homomorphisms: (M, p) −→ Tp M, h : (M, p) −→ (N, q) −→ dp h : Tp M −→ Tq N ; these mappings take a composition of morphisms to a composition of morphisms by (4.5) and idM to idTp M . On the other hand, the pull-back map h∗ on the cotangent spaces reverses compositions of morphisms by (4.6) and thus gives rise to a contravariant functor between the same two categories. 5 Immersions and Submanifolds Definition 5.1. Let M and N be smooth manifolds. (1) A smooth map f : M −→ N is an immersion if the differential dp f : Tp M −→ Tf (p) N is injective for every p ∈ M . (2) The manifold M is a submanifold of N if M ⊂ N , M has the subspace topology, and the inclusion map ι : M −→ N is an immersion. If M ⊂ N is a smooth submanifold and p ∈ M , the differential dp ι : Tp M −→ Tp N is an injective homomorphism. In such cases, we will identify Tp M with Im dp ι ⊂ Tp N via dp ι. Discrete subsets of points (with the unique smooth structure) and open subsets (with the induced smooth structure of Proposition 1.11) of a smooth manifold are submanifolds; see Exercise 25. If M and N are smooth manifolds, the horizontal and vertical slices Im ιq , Im ιp ⊂ M ×N of Example 2.5 are embedded submanifolds; see Exercise 26. On the other hand, Q ⊂ R does not admit a submanifold structure. If f : M −→ N is a diffeomorphism between smooth manifolds, then the differential dp f : Tp M −→ Tf (p) N (5.1) is an isomorphism for every p ∈ M . Thus, a diffeomorphism between two smooth manifolds is a bijective immersion. On the other hand, if f : M −→ N is an immersion, dim M ≤ dim N . If dim M = dim N and f : M −→ N is an immersion, then the differential (5.1) is an isomorphism for every p ∈ M . Corollary 4.9 then implies that f is a local diffeomorphism. Thus, a bijective immersion f : M −→ N between smooth manifolds of the same dimension is a diffeomorphism. The assumption that manifolds are second-countable topological spaces turns out to imply that a bijective immersion must be a map between manifolds of the same dimension; see Exercise 31. Thus, a bijective immersion is a diffeomorphism and vice versa. 24 Rn−m Nn Mm p U f f (p) ψ π Rm V ϕ = π◦ψ◦f Figure 1.8: An immersion pull-backs a subset of the coordinates on the target to a smooth chart on the domain A more interesting example of an immersion is the inclusion of Rm as the coordinate subspace Rm×0 into Rn , with m ≤ n. By Proposition 5.3 below, every immersion f : M −→ N locally (on M and N ) looks like the inclusion of Rm as Rm× 0 into Rn and every submanifold M ⊂ N locally (on N ) looks like Rm×0 ⊂ Rn . We will use the following lemma in the proof of Proposition 5.3. Lemma 5.2. Let f : M m −→ N n be a smooth map and p ∈ M . If the differential dp f is injective, there exist a neighborhood U of p in M and a smooth chart ψ = (y1 , . . . , yn ) : V −→ Rn around f (p) ∈ N such that ϕ = (y1 ◦f, . . . , ym ◦f ) : U −→ Rm is a smooth chart around p ∈ M . Since the differential dp f : Tp M −→ Tf (p) N is injective, its dual f ∗ = {dp f }∗ : Tf∗(p) N −→ Tp∗ M is surjective. Thus, if ψ = (y1 , . . . , yn ) : V −→ Rn is any smooth chart around f (p) ∈ N , then the set ∗ f df (p) yi = dp (yi ◦f ) i spans Tp∗ M (because the set {df (p) yi } is a basis for Tf∗(p) N ). By Corollary 4.11, a subset of {yi◦f }i determines a smooth chart around p on M . If this subset is different from {y1 ◦f, . . . , ym ◦f }, compose ψ with a diffeomorphism of Rn that switches the coordinates, sending the chosen coordinates (those in the subset) to the first m coordinates. The statement of Lemma 5.2 is illustrated in Figure 1.8. In summary, if dp f is injective, then m of the coordinates of a smooth chart around f (p) give rise to a smooth chart around p. By re-ordering the coordinates around f (p), it can be assumed that it is the first m coordinates that give rise to a smooth chart around p, which is then ϕ = π ◦ψ◦f , where π : Rn −→ Rm is the projection on the first m coordinates. In particular, π : ψ(f (U )) −→ ϕ(U ) ⊂ Rm is bijective; so the image of f (U ) ⊂ V ⊂ N under ψ is the graph of some function g : ϕ(U ) −→ Rn−m : ψ(f (U )) = (x, g(x)) : x ∈ ϕ(U ) . By construction, ψ(f (p′ )) = y1 (f (p′ )), . . . , yn (f (p′ )) = ϕ(p′ ), g(ϕ(p′ )) ∈ Rm ×Rn−m 25 ∀ p′ ∈ U ; so g = (ym+1 , . . . , yn ) ◦ f ◦ ϕ−1 . In the proof of the next proposition, we compose ψ with the diffeomorphism (x, y) −→ (x, y−g(x)) so that the image of f (U ) is shifted to Rm ×0. Proposition 5.3 (Slice Lemma). Let f : M m −→ N n be a smooth map and p ∈ M . If dp f is injective, there exist smooth charts ϕ : U −→ Rm ψ : V −→ Rn and around p ∈ M and f (p) ∈ N , respectively, such that the diagram U f // V ϕ ψ Rm // Rn commutes, where the bottom arrow is the natural inclusion of Rm as Rm×0, and f (U ) = ψ −1 (Rm×0). By Lemma 5.2, there exist a neighborhood U of p in M and a smooth chart ψ ′ = (y1 , . . . , yn ) : V ′ −→ Rn around f (p) ∈ N such that ϕ = π◦ψ ′ ◦f : U −→ Rm is a smooth chart around p ∈ M , where π : Rn −→ Rm is the projection on the first m coordinates as before. In particular, ϕ(U ) ⊂ Rm is an open subset and ψ ′ ◦ f = ϕ, g◦ϕ : U −→ Rm ×Rn−m, where g = (ym+1 , . . . , yn )◦f ◦ϕ−1 : ϕ(U ) −→ Rn−m ; this is a smooth function. Thus, the map Θ : ϕ(U )×Rn−m −→ ϕ(U )×Rn−m , (x, y) −→ x, y − g(x) , is smooth. It is clearly bijective, and J (Θ)(x,y) = Im 0 ∗ In−m ; so Θ is a diffeomorphism. Let V = ψ ′−1 (ϕ(U )×Rn−m ) and ψ = Θ◦ψ ′ : V −→ Rn . Since ϕ(U )×Rn−m is open in Rn , V is open in N . Since Θ is a diffeomorphism, ψ is a smooth chart on N . Since ψ ′ (V ′ ) and ϕ(U )×Rn−m contain ψ ′ (f (U )), f (U ) is contained in V . By definition, ψ ◦ f (p′ ) = Θ◦ψ ′ ◦f (p′ ) = Θ ϕ(p′ ), g(ϕ(p′ )) = ϕ(p′ ), g(ϕ(p′ )) − g(ϕ(p′ )) = ϕ(p′ ), 0 ∈ ϕ(U )×0 ∀ p′ ∈ U. Since ψ(f (U )) = ϕ(U ) = ψ(V )∩Rm ×0, f (U ) = ψ −1 (Rm ×0). Corollary 5.4. If M m ⊂ N n is a submanifold, for every p ∈ M there exists a smooth chart ψ ≡ (x1 , . . . , xn ) : V −→ Rn on N around p such that M ∩V = ψ −1 (Rm ×0) ≡ p′ ∈ V : xm+1 (p′ ) = xm+2 (p′ ) = . . . = xn (p′ ) = 0 and ψ : M ∩V −→ Rm ×0 = Rm is a smooth chart on M . 26 Rn−m Nn Mm p U f f (p) ψ V Rm ϕ = ψ◦f Figure 1.9: The local structure of immersions Let U be an open neighborhood of p in M and (V, ψ) a smooth chart on N around p = f (p) provided by Proposition 5.3 for the inclusion map f : M −→ N . Since M ⊂ N has the subspace topology, there exists W ⊂ V open so that U = M ∩ W ; the smooth chart (W, ψ|W ) then has the desired properties. According to this corollary, every smooth m-submanifold M of an n-manifold N locally (in a suitably chosen smooth chart) looks like the horizontal slice Rm ×0 ⊂ Rn . If p ∈ M lies in such a chart, n ∂ ∂ ∂ o Tp M = SpanR , ,..., ∂x1 p ∂x2 p ∂xm p = v ∈ Tp N : dp xm+1 (v) = dp xm+2 (v) = . . . = dp xn (v) = 0 . Proposition 5.3 completely describes the local structure of immersions, but says nothing about their global structure. Images of 3 different immersions of R into R2 are shown in Figure 1.10. Another type phenomena is illustrated by the injective immersion R −→ S 1 ×S 1 , s −→ eis , eiαs , (5.2) where α ∈ R−Q. The image of this immersion is a dense submanifold of S 1 ×S 1 . If ι : M −→ N is an injective map and h : X −→ N is any map such that h(X) ⊂ ι(M ), then there Figure 1.10: Images of some immersions R −→ R2 27 exists a unique map h0 : X −→ M so that the diagram h0 X | | | h M |>> ι // N commutes. If M , N , and X are topological spaces, ι is an embedding, and h is continuous, then h0 is also continuous [7, Theorem 7.2e]. An analogue of this property holds in the smooth category, as indicated by the next proposition. Proposition 5.5. Let ι : M −→ N be an injective immersion, h : X −→ N a smooth map such that h(X) ⊂ ι(M ), and h0 : X −→ M the unique map such that h = ι◦h0 . If h0 is continuous, then it is smooth; in particular, h0 is smooth if ι is an embedding (e.g. if M is a submanifold of N ). It is sufficient to show that every point q ∈ X has a neighborhood W on which h0 is smooth. By Proposition 5.3, there exist smooth charts ϕ : U −→ Rm ψ : V −→ Rn and around h0 (q) ∈ M and h(q) = ι(h0 (q)) ∈ N such that the diagram h0 W } } h } U }>> ϕ // Rm ψ // Rn ι // V m m commutes, where W = h−1 0 (U ) and the right-most arrow is the standard inclusion of R as R ×0 in Rn . Since h0 is continuous, W is open in X. Since h is smooth and ψ is a smooth chart on N , the map ψ◦h = ψ◦ι◦h0 = ϕ◦h0 , 0 : W −→ Rm ×Rn−m is smooth. Thus, the map ϕ ◦ h0 : W −→ Rm is also smooth. Since ϕ is a smooth chart on M containing the image of h0 |W , it follows that h0 |W is a smooth map. It is possible for the map h0 to be continuous even if ι : M −→ N is not an embedding (and even if the image of h is not contained in the image of any open subset of M on which ι is an embedding). This is in particular the case for the immersion (5.2), which satisfies the condition of the following definition. Definition 5.6. An injective immersion ι : M m −→ N n is regular if for every q ∈ N there exists a smooth chart ψ : V −→ Rm × Rn−m around q such that the image of every connected subset U ⊂ ι−1 (V ) under ψ is contained in ψ −1 (Rm ×y) for some y ∈ Rn−m (dependent on U ). Since the connected components of ι−1 (V ) are disjoint open subsets of M and each of them is mapped by ι to one of the horizontal slices ψ −1 (Rm×y), ι−1 (V ) ⊂ M is contained in at most countably many of the horizontal slices ψ −1 (Rm ×y). In particular, each of the connected components of ι−1 (V ) ⊂ M lies in one of these slices; see Figure 1.11. Corollary 5.7. If ι : M −→ N is a regular immersion, h : X −→ N is a smooth map such that h(X) ⊂ ι(M ), and h0 : X −→ M is the unique map such that h = ι◦h0 , then h0 is smooth. 28 Rn−m ι(M ) ∩ V Rm Figure 1.11: Image of a regular immersion ι(M ) in a smooth chart consists of horizontal slices In light of Proposition 5.5, it is sufficient to show that the map h0 is continuous. Let U be a connected open subset of M , x ∈ h−1 0 (U ), and p = h0 (x). We will show that there is an open subset W ⊂ X such that x ∈ W and h(W ) ⊂ ι(U ); since ι is injective, the latter implies that h0 (W ) ⊂ U and m n−m so W ⊂ h−1 0 (U ). Since ι is a regular immersion, there exists a smooth chart ψ : V −→ R ×R ′ −1 around h0 (p) = h(x) ∈ N such that the image of every connected subset U ⊂ ι (V ) under ψ is contained in ψ −1 (Rm×y) for some y ∈ Rn−m. Shrinking U and V and shifting ψ, it can be assumed that ι(U ) = ψ −1 (Rm ×0). Let W ⊂ h−1 (V ) be the connected component containing x ∈ N . Since h(W ) ⊂ ι(M )∩V is connected, h(W ) is contained in one of the horizontal slices ψ −1 (Rm ×y). Since h(x) ∈ ψ −1 (Rm ×0), we conclude that h(W ) ⊂ ψ −1 (Rm ×0) = ι(U ). On the other hand, h0 in Proposition 5.5 need not be continuous in general. For example, it is not continuous at h−1 (0) if ι and h are immersions described by the middle and right-most diagrams, respectively, in Figure 1.10. A similar example can be obtained from the left diagram in Figure 1.10 if all branches of the curve have infinite contact with the x-axis at the origin (ι and h can then differ by a “branch switch” at the origin). Corollary 5.8. Let N be a smooth manifold, M ⊂ N , and ι : M −→ N the inclusion map. (1) If TM is a topology on M , there exists at most one smooth structure FM on (M, TM ) with respect to which ι is an immersion. (2) If TM is the subspace topology on M and (M, TM ) admits a smooth structure FM with respect ′ admitting a smooth structure to which ι is an immersion, there exists no other topology TM ′ on M with respect to which ι is an immersion. FM The first statement of this corollary follows easily from Proposition 5.5. The second statement depends on manifolds being second-countable; its proof makes use of Exercise 31. Corollary 5.9. A topological subspace M ⊂ N admits a smooth structure with respect to which M is a submanifold of N if and only if for every p ∈ M there exists a neighborhood U of p in N such that the topological subspace M ∩U of N admits a smooth structure with respect to which M ∩U is a submanifold of N . By Corollary 5.8, the smooth structures on the overlaps of such open subsets must agree. The middle and right-most diagrams in Figure 1.10 are examples of a subset M of a smooth manifold N that admits two different manifold structures (M, TM , FM ), in different topologies, with respect to which the inclusion map ι : M −→ N is an embedding. In light of the second statement 29 of Proposition 5.8, this is only possible because M does not admit such a smooth structure in the subspace topology. On the other hand, if manifolds were not required to be second-countable, the discrete topology on R would provide a second manifold structure with respect to which the identity map R −→ R, with the target R having the standard manifold structure, would be an immersion. 6 Submersions and Submanifolds This section is in a sense dual to Section 5. It describes ways of constructing new immersions and submanifolds by studying properties of submersions (smooth maps with surjective differentials), rather than studying properties of immersions and submanifolds. While Section 5 exploits Corollary 4.11, this section makes use of Corollary 4.12, as well as of the Slice Lemma (Proposition 5.3). If M and N are smooth manifolds, the component projection maps π1 : M ×N −→ M, π2 : M ×N −→ N, are submersions; see Exercise 26. By the following lemma, every submersion locally has this form. Lemma 6.1. Let h : M m −→ Z k be a smooth map and p ∈ M . If the differential dp h is surjective, there exist smooth charts ϕ : U −→ Rm and ψ : V −→ Rk around p ∈ M and h(p) ∈ Z, respectively, such that the diagram U ϕ h V ψ // Rm // Rk commutes, where the right arrow is the natural projection map from Rm to Rk ×0 ⊂ Rm . Let ψ = (y1 , . . . , yk ) : V −→ Rk be a smooth chart on Z around f (p). Since the differential dp h is surjective, its dual map ∗ h∗ = dp h}∗ : Th(p) N −→ Tp∗ M ∗ N , it follows that the set is injective. Since {dh(p) yi } is a basis for Th(p) ∗ h dh(p) yi = dp (yi ◦h) is linearly independent in Tp∗ M . By Corollary 4.12, it can be extended to a smooth chart ϕ : y1 ◦h, . . . , yk ◦h, xk+1 , . . . , xm : U −→ Rk ×Rm−k on M , where U is a neighborhood of p in h−1 (V ). Lemma 6.1 can be seen as a counter-part of the Slice Lemma (Proposition 5.3). While an immersion locally looks like the inclusion Rm −→ Rm ×0 ⊂ Rn , 30 m ≤ n, U Rm−k ϕ M h π h(p) V ψ ψ(h(p)) Rk Z Figure 1.12: The local structure of submersions a submersion locally looks like the projection Rm −→ Rk = Rk ×0 ⊂ Rm , k ≤ m. Thus, an immersion can locally be represented by a horizontal slice in a smooth chart, while the pre-image of a point in the target of a submersion is locally a vertical slice (it is customary to present projections vertically, as in Figure 1.12). Corollary 6.2. Let h : M −→ Z be a smooth map and p ∈ M . If the differential dp h is surjective, there exist a neighborhood U of p in M and a smooth structure on the subspace h−1 (h(p))∩U of M so that h−1 (h(p))∩U is a submanifold of M and codimM h−1 (h(p))∩U ≡ dim M − dim h−1 (h(p))∩U = dim Z. If ψ : V −→ Rk and ϕ = (ψ◦h, φ) : U −→Rk ×Rm−k are smooth charts on Z around h(p) and on M around p, respectively, provided by Lemma 6.1, h−1 (h(p))∩U = {ψ◦h}−1 ψ(h(p)) ∩U = {π◦ϕ}−1 ψ(h(p)) = ϕ−1 ψ(h(p))×Rm−k . Since ϕ : U −→ ϕ(U ) is a homeomorphism, so is the map ϕ : h−1 (h(p))∩U −→ ψ(h(p))×Rm−k ∩ ϕ(U ) in the subspace topologies. Thus, φ : h−1 (h(p))∩U −→ Rm−k induces a smooth structure on h−1 (h(p))∩U ⊂ M in the subspace topology with respect to which the inclusion h−1 (h(p))∩U −→ M is an immersion because so is the inclusion ψ(h(p))×Rm−k −→ Rk ×Rm−k . Theorem 6.3 (Implicit Function Theorem for Manifolds). Let f : M −→ N be a smooth map and Y ⊂ N an embedded submanifold. If Tf (p) N = Im dp f + Tf (p) Y ∀ p ∈ f −1 (Y ), then f −1 (Y ) admits the structure of an embedded submanifold of M , codimM f −1 (Y ) = codimN Y, Tp f −1 (Y ) = {dp f }−1 Tf (p) Y ∀ p ∈ f −1 (Y ). 31 (6.1) In order to prove the first two statements, it is sufficient to show that for every p ∈ f −1 (Y ) there exists a neighborhood U of p in M such that f −1 (Y ) ∩ U admits the structure of an embedded submanifold of M of the claimed dimension; see Corollary 5.9. As provided by Corollary 5.4, let ψ : V −→ Rn be a smooth chart on N around f (p) ∈ Y such that Y ∩V = ψ −1 (Rl×0), where l = dim Y . Let π̃ : Rn −→ 0×Rn−l be the projection map and h = π̃◦ψ◦f : f −1 (V ) −→ V −→ Rn −→ Rn−l . Since Rl ×0 = π̃ −1 (0), Y ∩V = ψ −1 (π̃ −1 (0)) and f −1 (Y )∩f −1 (V ) = f −1 (Y ∩V ) = f −1 ψ −1 (π̃ −1 (0)) = h−1 (0). (6.2) On the other hand, by the chain rule (4.5) dp h = dψ(f (p)) π̃ ◦ df (p) ψ ◦ dp f : Tp M −→ Tf (p) N −→ Tψ(f (p)) Rn −→ T0 (0×Rn−l ). (6.3) The homomorphism dψ(f (p)) π̃ is onto, as is the homomorphism df (p) ψ. On the other hand, dψ(f (p)) π̃ ◦ df (p) ψ = df (p) (π̃ ◦ψ) : Tf (p) N −→ Tψ(f (p)) Rn −→ T0 (0×Rn−l ) by the chain rule (4.5) and thus vanishes on Tf (p) Y (since π̃ ◦ ψ maps Y to 0 in 0 × Rn−l ). So, by (6.1), the restriction dψ(f (p)) π̃◦df (p) ψ : Im dp f −→ T0 (0×Rn−l ) is onto, i.e. the homomorphism (6.3) is surjective. By Corollary 6.2 and (6.2), there exists a neighborhood U of p in f −1 (V ) such that f −1 (Y ) ∩ U = f −1 (Y ) ∩ f −1 (V ) ∩ U = h−1 (0) ∩ U admits the structure of an embedded submanifold of M of codimension l, as required. For the last statement, note that ∀ p ∈ f −1 (Y ), Tp f −1 (Y ) ⊂ {dp f }−1 Tf (p) Y since f (f −1 (Y )) ⊂ Y ; the opposite inclusion holds for dimensional reasons. Corollary 6.4. Let f : M −→ N be a smooth map and q ∈ N . If dp f : Tp M −→ Tq N is onto ∀ p ∈ f −1 (q), (6.4) then f −1 (q) admits the structure of an embedded submanifold of M of codimension equal to the dimension of N and Tp f −1 (q) = ker dp f : Tp M −→ Tq N ∀ p ∈ f −1 (q). This is just the Y = {q} case of Theorem 6.3. Example 6.5. Let f : Rm+1 −→ R be given by f (x) = |x|2 . This is a smooth map, and its differential at x ∈ Rm+1 with respect to the standard bases for Tx Rm+1 and Tf (x) R is J (f )x = 2x1 2x2 . . . 2xm+1 : Rm+1 −→ R. Thus, dx f is surjective if and only if x 6= 0, i.e. f (x) 6= 0. By Corollary 6.4, f −1 (q) with q 6= 0 then admits the structure of an embedded submanifold of Rm+1 and its codimension is 1 (so the √ dimension is m). This is indeed the case, since f −1 (q) is the sphere of radius q centered at the origin if q > 0 and the empty set (which is a smooth manifold of any dimension) if q < 0. If q = 0, f −1 (q) = {0}; this happens to be a smooth submanifold of Rm+1 , but of the wrong dimension. 32 Example 6.6. Corollary 6.4 can be used to show that the group SOn is a smooth submanifold of Matn×n R, while Un and SUn are smooth submanifolds of Matn×n C. For example, with SMatn R denoting the space of symmetric n×n real matrices, define f : Matn×n R −→ SMatn×n R, by f (A) = AAtr . Then, O(n) = f −1 (In ). It is then sufficient to show that the differential dA f is onto for all A ∈ O(n). Since f = f ◦RA for every A ∈ O(n), where the diffeomorphism RA : Matn×n R −→ Matn×n R is given by RA (B) = BA, it is sufficient to establish that dI f is surjective. This is done in Example 4.7. Corollary 6.7 (Implicit Function Theorem for Maps). Let f : X −→ M and g : Y −→ M be smooth maps. If Tf (x) M = Im dx f + Im dy g ∀ (x, y) ∈ X ×Y s.t. f (x) = g(y), (6.5) then the space X ×M Y ≡ (x, y) ∈ X ×Y : f (x) = g(y) admits the structure of an embedded submanifold of X ×Y of codimension equal to the dimension of M and T(p,q) X ×M Y = (v, w) ∈ Tp X ⊕Tq Y : dp f (v) = dq g(w) ∀ (p, q) ∈ X ×M Y under the identification of Exercise 26. Furthermore, the projection map π1 = πX : X ×M Y −→ X is injective (an immersion) if g : Y −→ M is injective (an immersion). This corollary is obtained by applying Theorem 6.3 to the smooth map h = (f, g) : X ×Y −→ M ×M. Its last statement immediately implies Warner’s Theorem 1.39. The commutative diagram X ×M Y π2 // Y g π1 X f // M is known as a fibered square. Corollary 6.8 (Implicit Function Theorem for Intersections). Let X, Y ⊂ M be embedded submanifolds. If Tp M = Tp X + Tp Y ∀ p ∈ X ∩Y, (6.6) then X ∩Y is a smooth submanifold of X, Y , and M , dim X ∩Y = dim X + dim Y − dim M, Tp (X ∩Y ) = Tp X ∩ Tp Y ⊂ Tp M ∀ p ∈ X ∩Y. This corollary is a special case of Corollary 6.7. 33 Remark 6.9. Submanifolds X, Y ⊂ M satisfying (6.6) are said to be transverse (in M ); this is written as X ⊤ ∩ Y or X ⊤ ∩ M Y to be specific. For example, two distinct lines in the plane are transverse, but two intersecting lines in R3 are not. Similarly, smooth maps f : X −→ M and g : Y −→ M satisfying (6.5) are called transverse; this is written as f⊤ ∩g or f⊤ ∩ M g. If f : M −→ N satisfies (6.1) with respect to a submanifold Y ⊂ N , f is said to transverse to Y ; this is written as f⊤ ∩ Y or f⊤ ∩ N Y . Finally, if f : M −→ N satisfies (6.4) with respect to q ∈ N , q is said to be a regular value of f . By Corollary 6.4, the pre-image of a regular value is a smooth submanifold in the domain of codimension equal to the dimension of the target. By Sard’s Theorem [4, §2], the set of a regular values is dense in the target (in fact, its complement is a set of measure 0); so the pre-images of most points in the target of a smooth map are smooth submanifolds of the domain, though in some cases they may all be empty (e.g. if the dimension of the domain is lower than the dimension of the target). The standard version of the Implicit Function Theorem for Rm , Corollary 6.10 below, says that under certain conditions a system of k equations in m variables has a locally smooth (m − k)dimensional space of solutions which can be described as a graph of a function g : Rm−k −→ Rk . It is normally obtained as an application of the Inverse Function Theorem for Rm , Theorem 4.8 above. It can also be deduced from the proof of Lemma 6.1 and by itself implies Corollary 6.2. Corollary 6.10 (Implicit Function Theorem for Rm ). Let U ⊂ Rm−k ×Rk be an open subset and f : U −→ Rk a smooth function. If (x0 , y0 ) ∈ f −1 (0) is such that the right k × k submatrix of m−k and J (f )(x0 ,y0 ) , ∂f ∂y |(x0 ,y0 ) , is non-singular, then there exist open neighborhoods V of x0 in R W of y0 in Rk and a smooth function g : V −→ W such that f −1 (0) ∩ V ×W = (x, g(x)) : x ∈ V . Exercises 1. Show that the collection (1.1) is indeed a smooth structure on M , according to Definition 1.3. 2. Show that the maps ϕ± : U± −→ Rm described after Example 1.7 are indeed charts on S m and the overlap map between them is m m ϕ+ ◦ϕ−1 − : ϕ− (U+ ∩U− ) = R −0 −→ ϕ+ (U+ ∩U− ) = R −0, x −→ x . |x|2 3. Show that the map ϕ1/2 in Example 1.8 is well-defined and is indeed a homeomorphism. 4. With notation as in Example 1.10, show that (a) the map S 2n+1 /S 1 −→ (Cn+1 −0)/C∗ induced by inclusions S 2n+1 −→ C2n+1 and S 1 −→ C∗ is a homeomorphism with respect to the quotient topologies; (b) the quotient topological space, CP n , is a compact topological 2n-manifold which admits a structure of a complex (in fact, algebraic) n-manifold, i.e. it can be covered by charts whose n overlap maps, ϕα ◦ ϕ−1 β , are holomorphic maps between open subsets of C (and rational n functions on C ); (c) CP n contains Cn (with its complex structure) as a dense open subset. 34 5. Let V and W be finite-dimensional vector spaces with the canonical smooth structures of Example 1.5. Show that the canonical smooth structure on the vector space V ⊕W = V ×W is the same as the product smooth structure. 6. Show that a composition of two smooth maps (local diffeomorphisms, diffeomorphisms) is again smooth (a local diffeomorphism, a diffeomorphism). 7. Let f : M −→ N be a map between smooth manifolds. Show that f is a smooth map if and only if for every smooth function h : N −→ R the function h◦f : M −→ R is also smooth. 8. Verify Lemma 2.2. 9. Let V be a finite-dimensional vector space with the canonical smooth structure of Example 1.5. Show that the vector space operations, V × V −→ V, R × V −→ V, (v1 , v2 ) −→ v1 + v2 , (r, v) −→ rv , are smooth maps. 10. Let V1 , V2 , W be finite-dimensional vector spaces with their canonical smooth structures of Example 1.5. Show that any bilinear map · : V1 ×V2 −→ W, v1 ⊗v2 −→ v1 ·v2 , is smooth. 11. Show that the two smooth structures F and F ′ on R1 in Example 1.6 are not the same, but (R1 , F) and (R1 , F ′ ) are diffeomorphic smooth manifolds. 12. Let S 1 ⊂ C and M B be the unit circle and the infinite Mobius band with the smooth structures of Examples 1.7 and 1.8, respectively. Show that the map MB = [0, 1] × R / ∼−→ S 1 , [s, t] −→ e2πis , is well-defined and smooth. 13. Let (M, F) be a smooth m-manifold and U ⊂ M an open subset. Show that F|U is the unique smooth structure on the topological subspace U of M satisfying either of the following two properties: (SSM1) the inclusion map ι : U −→ M is a local diffeomorphism; (SSM2) if N is a smooth manifold, a continuous map f : N −→ U is smooth if and only if the map ι◦f : N −→ M is smooth. 14. Let (M, FM ) and (N, FN ) be smooth manifolds and FM ×N the product smooth structure on M ×N of Proposition 1.11. Show that FM ×N is the unique smooth structure on the product topological space M ×N satisfying either of the following two properties: (PSM1) the slice inclusion maps ιq : M −→ M × N , with q ∈ N , and ιp : M −→ M × N , with p ∈ M , and the projection maps πM , πN : M ×N −→ M, N are smooth; 35 (PSM2) if X is a smooth manifold, continuous maps f : X −→ M and g : X −→ N are smooth if and only if the map f ×g : X −→ M ×N is smooth. 15. Verify Lemmas 2.6 and Corollary 2.7. 16. Verify Proposition 2.8. 17. Show that the actions (2.4), (2.5), and (1.3) satisfy the assumptions of Proposition 2.8 and that the quotient smooth structures on S 1 = R/Z, MB = (R×R)/Z, and RP n = S n /Z2 , are the same as the smooth structures of Examples 1.7, 1.8, and 1.9, respectively. 18. Verify that the addition and product operations on F̃p described after Definition 3.1 are welldefined and make F̃p into an R-algebra with valuation map evp . 19. Deduce (3.7) from Lemma 3.5. 20. Verify that the map (3.9) is well-defined and is indeed an R-algebra homomorphism. 21. Verify that the differential dp h of a smooth map h : M −→ N , as defined in (4.1), is indeed well-defined. In other words, dp h(v) is a derivation on F̃h(p) for all v ∈ Tp M . Show that dp h : Tp M −→ Th(p) N is a vector-space homomorphism. 22. Let M be a smooth connected manifold and f : M −→ N a smooth map. Show that dp f = 0 for all p ∈ M if and only if f is a constant map. 23. Let M be a smooth manifold, V a finite-dimensional vector space with the canonical smooth structure of Example 1.5, and f, g : M −→ V smooth maps. Show that dp (f +g) = dp f + dp g : Tp M −→ V ∀ p ∈ M, under the canonical identifications Tf (p) V, Tg(p) , Tf (p)+g(p) V = V of Example 3.8. 24. Let f, g : M −→ R be smooth maps. Show that dp (f g) = f (p)dp g + g(p)dp f : Tp M −→ R ∀ p ∈ M. More generally, suppose V1 , V2 , W are finite-dimensional vector spaces with their canonical smooth structures of Example 1.5, · : V1 ⊗V2 −→ W, v1 ⊗v2 −→ v1 ·v2 , is a bilinear map, and f1 : M −→ V1 and f2 : M −→ V2 are smooth maps. Show that dp (f1 · f2 ) = f1 (p) · dp f2 + dp f1 · f2 (p) : Tp M −→ W ∀ p ∈ M, under the canonical identifications Tf1 (p) V1 = V1 , Tf2 (p) V2 = V2 , and Tf1 (p)·f2 (p) W = W of Example 3.8. 36 25. Let (M, F) be a smooth manifold, U ⊂ M an open subset with the smooth structure induced from M as in Proposition 1.11, and ι : U −→ M the inclusion map. Show that the differential dp ι : Tp U −→ Tp M is an isomorphism for all p ∈ U . 26. Let (M, FM ) and (N, FN ) be smooth manifolds and M ×N their Cartesian product with the product smooth structure of Proposition 1.11. With notation as in Example 2.5, show that the homomorphisms dp ιq +dq ιp : Tp M ⊕ Tq N −→ T(p,q) (M ×N ), d(p,q) π1 ⊕d(p,q) π2 : T(p,q) (M ×N ) −→ Tp M ⊕ Tq N, (v1 , v2 ) −→ dp ιq (v1 ) + dq ιp (v2 ) w −→ d(p,q) π1 (w), d(p,q) π2 (w) , are isomorphisms and mutual inverses for all p ∈ M and q ∈ N . 27. Let M be a non-empty compact m-manifold. Show that there exists no immersion f : M −→ Rm . 28. Show that there exists no immersion f : S 1 ×S 1 −→ RP 2 . 29. Let M be a smooth manifold and p ∈ M a fixed point of a smooth map f : M −→ M , i.e. f (p) = p. Show that if all eigenvalues of the linear transformation dp f : Tp M −→ Tp M are different from 1 (so dp f (v) 6= v for all v ∈ Tp M −0), then p is an isolated fixed point (has a neighborhood that contains no other fixed point). 30. Let M be an embedded submanifold in a smooth manifold N and ι : M −→ N the inclusion map. Show that for every p ∈ M the image of the differential dp ι : Tp M −→ Tp N is the subspace of Tp N consisting of the vectors α′ (0), where α : (−ǫ, ǫ) −→ N is a smooth map such that Im α ⊂ M and α(0) = p. 31. Show that a bijective immersion f : M −→ N between two smooth manifolds is a diffeomorphism. Hint: you’ll need to use that M is second-countable, along with either (i) if f : U −→ Rn is a smooth map from an open subset of Rm with m < n, the measure of f (U ) ⊂ Rn is 0; (ii) the Slice Lemma (Proposition 5.3) and the Baire Category Theorem [7, Theorem 7.2]. 32. Show that the map (5.2) is an injective immersion and that its image is dense in S 1 ×S 1 . 33. Verify Corollary 5.8. 34. Show that the smooth structures on S m of Example 6.5 and Exercise 2 are the same. 35. Show that the topological subspace {(x, y) ∈ R2 : x3 +xy+y 3 = 1 of R2 is a smooth curve (i.e. admits a natural structure of smooth 1-manifold with respect to which it is a submanifold of R2 ). 37 36. (a) For what values of t ∈ R, is the subspace (x1 , . . . , xn+1 ) ∈ Rn+1 : x21 +. . .+x2n −x2n+1 = t a smooth embedded submanifold of Rn+1 ? (b) For such values of t, determine the diffeomorphism type of this submanifold (i.e. show that it is diffeomorphic to something rather standard). Hint: Draw some pictures. 37. Show that the special unitary group SUn = A ∈ Matn C : Ātr A = In , det A = 1 is a smooth compact manifold. What is its dimension? 38. Verify Corollary 6.7. 39. With notation as in Corollary 6.7, show that every pair of continuous maps p : Z −→ X and q : Z −→ Y such that f ◦p = g◦q factors through a unique continuous map r : Z −→ X ×M Y , Z 5THTTT 55 H TTTT q 55 H rH TTTTT TTTT 55 H 55 ## π2 TTT)) // Y p 55X ×M Y 55 5π5 1 g 5 f // M X and that X×MY is the unique (up to homeomorphism) topological space possessing this property for all (p, q) as above. If in addition the assumption (6.5) holds and p and q are smooth, then r is also smooth, and X ×M Y is the unique (up to diffeomorphism) smooth manifold possessing this property for all (p, q) as above. 40. Verify Corollary 6.8. 41. Let M be a smooth manifold and p ∈ M a fixed point of a smooth map f : M −→ M , i.e. f (p) = p. Show that if all eigenvalues of the linear transformation dp f : Tp M −→ Tp M are different from 1 (so dp f (v) 6= v for all v ∈ Tp M −0), then p is an isolated fixed point (has a neighborhood that contains no other fixed point). 42. Deduce Corollary 6.10 from the proof of Lemma 6.1 and Corollary 6.2 from Corollary 6.10. 38 Chapter 2 Smooth Vector Bundles 7 Definitions and Examples A (smooth) real vector bundle V of rank k over a smooth manifold M is a smoothly varying family of k-dimensional real vector spaces which is locally trivial. Formally, it is a triple (M, V, π), where M and V are smooth manifolds and π : V −→ M is a surjective submersion. For each p ∈ M , the fiber Vp ≡ π −1 (p) of V over p is a real k-dimensional vector space; see Figure 2.1. The vector-space structures in Vp vary smoothly with p ∈ M . This means that the scalar multiplication map R × V −→ V, (c, v) −→ c · v, and the addition map V ×M V ≡ (v1 , v2 ) ∈ V ×V : π(v1 ) = π(v2 ) ∈ M −→ V, (7.1) (v1 , v2 ) −→ v1 +v2 , (7.2) are smooth. Note that we can add v1 , v2 ∈ V only if they lie in the same fiber over M , i.e. π(v1 ) = π(v2 ) ⇐⇒ (v1 , v2 ) ∈ V ×M V. The space V ×M V is a smooth submanifold of V ×V by the Implicit Function Theorem for Maps (Corollary 6.7). The local triviality condition means that for every point p ∈ M there exist a neighborhood U of p in M and a diffeomorphism h : V |U ≡ π −1 (U ) −→ U ×Rk such that h takes every fiber of π to the corresponding fiber of the projection map π1 : U×Rk −→ U , i.e. π1 ◦h = π on V |U so that the diagram h V |U ≡ π −1 (U ) ≈ LLL LLL π LLL LL%% U // U ×Rk y yy yy y y π1 ||yy commutes, and the restriction of h to each fiber is linear: h(c1 v1 +c2 v2 ) = c1 h(v1 ) + c2 h(v2 ) ∈ x × Rk 39 ∀ c1 , c2 ∈ R, v1 , v2 ∈ Vx , x ∈ U. Vp V π M p Figure 2.1: Fibers of a vector-bundle projection map are vector spaces of the same rank. These conditions imply that the restriction of h to each fiber Vx of π is an isomorphism of vector spaces. In summary, V locally (and not just pointwise) looks like bundles of Rk ’s over open sets in M glued together. This is in a sense analogous to an m-manifold being open subsets of Rm glued together in a nice way. Here is a formal definition. Definition 7.1. A real vector bundle of rank k is a tuple (M, V, π, ·, +) such that (RVB1) M and V are smooth manifolds and π : V −→ M is a smooth map; (RVB2) · : R×V −→ V is a map s.t. π(c·v) = π(v) for all (c, v) ∈ R×V ; (RVB3) + : V ×M V −→ V is a map s.t. π(v1 +v2 ) = π(v1 ) = π(v2 ) for all (v1 , v2 ) ∈ V ×M V ; (RVB4) for every point p ∈ M there exist a neighborhood U of p in M and a diffeomorphism h : V |U −→ U ×Rk such that (RVB4-a) π1 ◦h = π on V |U and (RVB4-b) the map h|Vx : Vx −→ x×Rk is an isomorphism of vector spaces for all x ∈ U . The spaces M and V are called the base and the total space of the vector bundle (M, V, π). It is customary to call π : V −→ M a vector bundle and V a vector bundle over M . If M is an mmanifold and V −→ M is a real vector bundle of rank k, then V is an (m+k)-manifold. Its smooth charts are obtained by restricting the trivialization maps h for V , as above, to small coordinate charts in M . Example 7.2. If M is a smooth manifold and k is a nonnegative integer, then π1 : M ×Rk −→ M is a real vector bundle of rank k over M . It is called the trivial rank k real vector bundle over M and denoted π : τkR −→ M or simply π : τk −→ M if there is no ambiguity. Example 7.3. Let M = S 1 be the unit circle and V = MB the infinite Mobius band of Example 1.8. With notation as in Example 1.8, the map [s, t] −→ e2πis , π : V −→ M, defines a real line bundle (i.e. rank 1 bundle) over S 1 . Trivializations of this vector bundle can be constructed as follows. With U± = S 1 −{±1}, let [s, t] −→ e2πis , t ; h+ : V |U+ −→ U+ ×R, ( (e2πis , t), if s ∈ (1/2, 1]; [s, t] −→ h− : V |U− −→ U− ×R, 2πis (e , −t), if s ∈ [0, 1/2). 40 Both maps are diffeomorphisms, with respect to the smooth structures of Example 1.8 on MB and of Example 1.7 on S 1 . Furthermore, π1 ◦h± = π and the restriction of h± to each fiber of π is a linear map to R. Example 7.4. Let RP n be the real projective space of dimension n described in Example 1.9 and γn = (ℓ, v) ∈ RP n ×Rn+1 : v ∈ ℓ , where ℓ ⊂ Rn+1 denotes a one-dimensional linear subspace. If Ui ⊂ RP n is as in Example 1.9, the map hi : γn ∩ Ui ×Rn+1 −→ Ui ×R, ℓ, (v0 , . . . , vn ) −→ (ℓ, vi ), is a homeomorphism. The overlap maps, hi ◦h−1 j : Ui ∩Uj × R −→ Ui ∩Uj × R, (ℓ, c) −→ ℓ, (Xi /Xj )c , are smooth. By Lemma 2.6, the collection {(γn ∩ Ui × Rn+1 , hi )} of generalized smooth charts then induces a smooth structure on the topological subspace γn ⊂ RP n ×Rn+1 . With this smooth structure, γn is an embedded submanifold of RP n×Rn+1 and the projection on the first component, π = π1 : γn −→ RP n , defines a smooth real line bundle. The fiber over a point ℓ ∈ RP n is the one-dimensional subspace ℓ of Rn+1 ! For this reason, γn is called the tautological line bundle over RP n . Note that γ1 −→ S 1 is the infinite Mobius band of Example 7.3. Example 7.5. If M is a smooth m-manifold, let G TM = Tp M, π : T M −→ M, p∈M If ϕα : Uα −→ Rm is a smooth chart on M , let ϕ̃α : T M |Uα ≡ π −1 (Uα ) −→ Uα × Rm , π(v) = p if v ∈ Tp M. ϕ̃α (v) = π(v), dπ(v) ϕα v . (7.3) If ϕβ : Uβ −→ Rm is another smooth chart, the overlap map m m ϕ̃α ◦ ϕ̃−1 β : Uα ∩Uβ × R −→ Uα ∩Uβ × R is a smooth map between open subsets of R2m . By Corollary 2.7, the collection of generalized smooth charts −1 (π (Uα ), ϕ̃α ) : (Uα , ϕα ) ∈ FM , where FM is the smooth structure of M , then induces a manifold structure on the set T M . With this smooth structure on T M , the projection π : T M −→ M defines a smooth real vector bundle of rank m, called the tangent bundle of M . Definition 7.6. A complex vector bundle of rank k is a tuple (M, V, π, ·, +) such that (CVB1) M and V are smooth manifolds and π : V −→ M is a smooth map; (CVB2) · : C×V −→ V is a map s.t. π(c·v) = π(v) for all (c, v) ∈ C×V ; 41 (CVB3) + : V ×M V −→ V is a map s.t. π(v1 +v2 ) = π(v1 ) = π(v2 ) for all (v1 , v2 ) ∈ V ×M V ; (CVB4) for every point p ∈ M there exists a neighborhood U of p in M and a diffeomorphism h : V |U −→ U ×Ck such that (CVB4-a) π1 ◦h = π on V |U and (CVB4-b) the map h|Vx : Vx −→ x×Ck is an isomorphism of complex vector spaces for all x∈U. Similarly to a real vector bundle, a complex vector bundle over M locally looks like bundles of Ck ’s over open sets in M glued together. If M is an m-manifold and V −→ M is a complex vector bundle of rank k, then V is an (m+2k)-manifold. A complex vector bundle of rank k is also a real vector bundle of rank 2k, but a real vector bundle of rank 2k need not in general admit a complex structure. Example 7.7. If M is a smooth manifold and k is a nonnegative integer, then π1 : M ×Ck −→ M is a complex vector bundle of rank k over M . It is called the trivial rank-k complex vector bundle over M and denoted π : τkC −→ M or simply π : τk −→ M if there is no ambiguity. Example 7.8. Let CP n be the complex projective space of dimension n described in Example 1.10 and γn = (ℓ, v) ∈ CP n ×Cn+1 : v ∈ ℓ . The projection π : γn −→ CP n defines a smooth complex line bundle. The fiber over a point ℓ ∈ CP n is the one-dimensional complex subspace ℓ of Cn+1 . For this reason, γn is called the tautological line bundle over CP n . Example 7.9. If M is a complex m-manifold, the tangent bundle T M of M is a complex vector bundle of rank m over M . 8 Sections and Homomorphisms Definition 8.1. (1) A (smooth) section of a (real or complex) vector bundle π : V −→ M is a (smooth) map s : M −→ V such that π◦s = idM , i.e. s(x) ∈ Vx for all x ∈ M . (2) A vector field on a smooth manifold is a section of the tangent bundle T M −→ M . If π : V = M × Rk −→ M is the trivial bundle of rank k, a section of π is a map s : M −→ V of the form s = (idM , f ) : M −→ M ×Rk for some map f : M −→ Rk . This section is smooth if and only if f is a smooth map. Thus, a (smooth) section of the trivial vector bundle of rank k over M is essentially a (smooth) map M −→ Rk . If s is a smooth section, then s(M ) is an embedded submanifold of V : the injectivity of s and ds is immediate from π◦s = idM , while the embedding property follows from the continuity of π. Every fiber Vx of V is a vector space and thus has a distinguished element, the zero vector in Vx , which 42 V s0 (M ) ≈ M s(M ) ≈ M Figure 2.2: The image of a vector-bundle section is an embedded submanifold of the total space. we denote by 0x . It follows that every vector bundle admits a canonical section, called the zero section, s0 (x) = (x, 0x ) ∈ Vx . This section is smooth, since on a trivialization of V over an open subset U of M it is given by the inclusion of U as U ×0 into U ×Rk or U ×Ck . Thus, M can be thought of as sitting inside of V as the zero section, which is a deformation retract of V ; see Figure 2.2. If s : M −→ V is a section of a vector bundle V −→ M and h : V |U −→ U ×Rk is a trivialization of V over an open subset U ⊂ M , then h ◦ s = (idU , sh ) : U −→ U ×Rk (8.1) for some sh : U −→ Rk . Since the trivializations h cover V and each trivialization h is a diffeomorphism, a section s : M −→ V is smooth if and only if the induced functions sh : U −→ Rk are smooth in all trivializations h : V |U −→ U ×Rk of V . Every trivialization h : V |U −→ U × Rk of a vector bundle V −→ M over an open subset U ⊂ M corresponds to a k-tuple (s1 , . . . , sk ) of smooth sections of V over U such that the set {si (x)}i forms a basis for Vx ≡ π −1 (x) for all x ∈ U . Let e1 , . . . , ek be the standard coordinate vectors in Rk . If h : V |U −→ U ×Rk is a trivialization of V , then each section si = h−1 ◦ (idU , ei ) : U −→ V |U , si (x) = h−1 (x, ei ), is smooth. Since {ei } is a basis for Rk and h : Vx −→ x×Rk is a vector-space isomorphism, {si (x)}i is a basis for Vx for all x ∈ U . Conversely, if s1 , . . . , sk : U −→ V |U are smooth sections such that {si (x)}i is a basis for Vx for all x ∈ U , then the map ψ : U × Rk −→ V |U , (x, c1 , . . . , ck ) −→ c1 s1 (x) + . . . + ck sk (x), (8.2) is a diffeomorphism commuting with the projection maps; its inverse, h = ψ −1 , is thus a trivialization of V over U . If in addition s : M −→ V is any bundle section and sh ≡ sh,1 , . . . , sh,k ) : U −→ Rk is as in (8.1), then s(x) = h−1 x, sh,1 (x), . . . , sh,k (x) = sh,1 (x)s1 (x) + . . . + sh,k (x)sk (x) ∀ x ∈ U. Thus, a bundle section s : M −→ V is smooth if and only if for every open subset U ⊂ M and a k-tuple of smooth sections s1 , . . . , sk : U −→ V |U such that {si (x)}i is a basis for Vx for all x ∈ U the coefficient functions c1 , . . . , ck : U −→ R, s(x) ≡ c1 (x)s1 (x) + . . . + ck (x)sk (x) ∀ x ∈ U, 43 are smooth. For example, let π : V = T M −→ M be the tangent bundle of a smooth m-manifold M . If ϕ̃α is a trivialization of T M over Uα ⊂ M as in (7.3), ∂ −1 si (x) ≡ ϕ̃α (x, ei ) = ∀ x ∈ Uα ∂xi x is the i-th coordinate vector field. Thus, a vector field X : M −→ T M is smooth if and only if for every smooth chart ϕα = (x1 , . . . , xm ) : Uα −→ Rm the coefficient functions ∂ ∂ + . . . + cm (p) ∀ p ∈ U, c1 , . . . , cm : U −→ R, X(p) ≡ c1 (p) ∂x1 p ∂xm p are smooth. If X : M −→ T M is a vector field on M and p ∈ M , sometimes it will be convenient to denote the value X(p) ∈ Tp M of X at p by Xp . If in addition f ∈ C ∞ (M ), define Xf : M −→ R {Xf }(p) = Xp (f ) by ∀ p ∈ M. A vector field X on M is smooth if and only if Xf ∈ C ∞(M ) for every f ∈ C ∞ (M ). The set of all smooth sections of a vector bundle π : V −→ M is denoted by Γ(M ; V ). This is naturally a module over the ring C ∞ (M ) of smooth functions on M , since f s ∈ Γ(M ; V ) whenever f ∈ C ∞ (M ) and s ∈ Γ(M ; V ). We will denote the set Γ(M ; T M ) of smooth vector fields on M by VF(M ). It carries a canonical structure of Lie algebra over R, with the Lie bracket defined by [·, ·] : VF(M ) × VF(M ) −→ VF(M ), ∀ p ∈ M, f ∈ C ∞ (U ), U ⊂ M open, p ∈ U ; [X, Y ]p (f ) = Xp (Y f ) − Yp (Xf ) (8.3) see Exercise 5. Definition 8.2. (1) Suppose π : V −→ M and π ′ : V ′ −→ N are real (or complex) vector bundles. A (smooth) map f˜: V −→ V ′ is a (smooth) vector-bundle homomorphism if f˜ descends to a map f : M −→ N , i.e. the diagram f˜ V // V ′ π (8.4) π′ f M // N commutes, and the restriction f˜: Vx −→ Vf (x) is linear (or C-linear, respectively) for all x ∈ M . (2) If π : V −→ M and π ′ : V ′ −→ M are vector bundles, a smooth vector-bundle homomorphism f˜: V −→ V ′ is an isomorphism of vector bundles if π ′ ◦ f˜= π, i.e. the diagram f˜ VA A AA AA π AA M // V ′ | | || ||π′ | ~~| (8.5) commutes, and f˜ is a diffeomorphism (or equivalently, its restriction to each fiber is an isomorphism of vector spaces). If such an isomorphism exists, then V and V ′ are said to be isomorphic vector bundles. 44 Let f˜: V −→ V ′ be a vector-bundle homomorphism between vector bundles over the same space M that covers idM as in (8.5). If h : V |U −→ U ×Rk and h′ : V ′ |U −→ U ×Rk ′ are trivializations of V and V ′ over the same open subset U ⊂ M , then there exists f˜h′ h : U −→ Matk′ ×k R s.t. h′ ◦ f˜ ◦ h−1 (x, v) = x, f˜h′ h (x)v ∀ x ∈ U, v ∈ Rk . (8.6) Since the trivializations h and h′ are diffeomorphisms that cover V and V ′ , respectively, a vectorbundle homomorphism as in (8.5) is smooth if and only if the induced function f˜hh′ : U −→ Matk′ ×k R ′ is smooth for every pair, h : V |U −→ U ×Rk and h′ : V ′ |U −→ U ×Rk , of trivializations of V and V ′ over U . Example 8.3. The tangent bundle π : T Rn −→ Rn of Rn is canonically trivial. The map T Rn −→ Rn × Rn , v −→ π(v); v(π1 ), . . . , v(πn ) , where πi : Rn −→ R are the component projection maps, is a vector-bundle isomorphism. Lemma 8.4. The real line bundle V −→ S 1 given by the infinite Mobius band of Example 7.3 is not isomorphic to the trivial line bundle S 1 ×R −→ S 1 . Proof: In fact, (V, S 1 ) is not even homeomorphic to (S 1 ×R, S 1 ). Since S 1 ×R − s0 (S 1 ) ≡ S 1 ×R − S 1 ×0 = S 1 ×R− ⊔ S 1 ×R+ , the space S 1 × R − S 1 is not connected. On the other hand, V − s0 (S 1 ) is connected. If M B is the standard Mobius Band and S 1 ⊂ M B is the central circle, M B−S 1 is a deformation retract of V −S 1 . On the other hand, the boundary of M B has only one connected component (this is the primary feature of M B) and is a deformation retract of M B −S 1 . Thus, V −S 1 is connected as well. Lemma 8.5. If π : V −→ M is a real (or complex) vector bundle of rank k, V is isomorphic to the trivial real (or complex) vector bundle of rank k over M if and only if V admits k sections s1 , . . . , sk such that the vectors s1 (x), . . . , sk (x) are linearly independent over R (or over C, respectively) in Vx for all x ∈ M . Proof: We consider the real case; the proof in the complex case is nearly identical. (1) Suppose ψ : M ×Rk −→ V is an isomorphism of vector bundles over M . Let e1 , . . . , ek be the standard coordinate vectors in Rk . Define sections s1 , . . . , sk of V over M by ∀ i = 1, . . . , k, x ∈ M. si (x) = ψ x, ei Since the maps x −→ (x, ei ) are sections of M ×Rk over M and ψ is a bundle homomorphism, the maps si are sections of V . Since the vectors (x, ei ) are linearly independent in x × Rk and ψ is an isomorphism on every fiber, the vectors s1 (x), . . . , sk (x) are linearly independent in Vx for all x ∈ M , as needed. 45 (2) Suppose s1 , . . . , sk are sections of V such that the vectors s1 (x), . . . , sk (x) are linearly independent in Vx for all x ∈ M . Define the map ψ : M ×Rk −→ V by ψ(x, c1 , . . . , ck ) = c1 s1 (x) + . . . + ck sk (x) ∈ Vx . Since the sections s1 , . . . , sk and the vector space operations on V are smooth, the map h is smooth. It is immediate that π(ψ(x, c)) = x and the restriction of ψ to x×Rk is linear; thus, ψ is a vector-bundle homomorphism. Since the vectors s1 (x), . . . , sk (x) are linearly independent in Vx , the homomorphism ψ is injective and thus an isomorphism on every fiber. We conclude that ψ is an isomorphism between vector bundles over M . 9 Transition Data Suppose π : V −→ M is a real vector bundle of rank S k. By Definition 7.1, there exists a collection {(Uα , hα )}α∈A of trivializations for V such that α∈A Uα = M . Since (Uα , hα ) is a trivialization for V , hα : V |Uα −→ Uα ×Rk is a diffeomorphism such that π1◦hα = π and the restriction hα : Vx −→ x×Rk is linear for all x ∈ Uα . Thus, for all α, β ∈ A, k k hαβ ≡ hα ◦h−1 β : Uα ∩ Uβ × R −→ Uα ∩Uβ × R is a diffeomorphism such that π1 ◦hαβ = π1 , i.e. hαβ maps x×Rk to x×Rk , and the restriction of hαβ to x×Rk defines an isomorphism of x×Rk with itself. Such a diffeomorphism must be given by (x, v) −→ x, gαβ (x)v ∀ v ∈ Rk , for a unique element gαβ (x) ∈ GLk R (the general linear group of Rk ). The map hαβ is then given by hαβ (x, v) = x, gαβ (x)v ∀ x ∈ Uα ∩Uβ , v ∈ Rk , and is completely determined by the map gαβ : Uα ∩Uβ −→ GLk R (and gαβ is determined by hαβ ). Since hαβ is smooth, so is gαβ . Example 9.1. Let π : V −→ S 1 be the Mobius band line bundle of Example 7.3. the pair of trivializations described in Example 7.3, then ( (p, v), −1 h− ◦h+ : U+ ∩U− × R −→ U+ ∩U− × R, (p, v) −→ p, g−+ (p)v = (p, −v), ( −1, if where g−+ : U+ ∩U− = S 1 −{±1} −→ GL1 R = R∗ , g−+ (p) = 1, if If {(U± , h± )} is if Im p < 0, if Im p > 0, Im p > 0; Im p < 0. In this case, the transition maps gαβ are locally constant, which is rarely the case. Suppose {(Uα , hα )}α∈A is a collection of trivializations of a rank k vector bundle π : V −→ M covering M . Any (smooth) section s : M −→ V of π determines a collection of (smooth) maps {sα : Uα −→ Rk }α∈A such that hα ◦s(x) = x, sα (x) ∀ x ∈ Uα =⇒ sα (x) = gαβ (x)sβ (x) ∀ x ∈ Uα ∩Uβ , α, β ∈ A, (9.1) 46 where {gαβ }α,β∈A is the transition data for the collection of trivializations {hα }α∈A of V . Conversely, a collection of (smooth) maps {sα : Uα −→ Rk }α∈A satisfying the second condition in (9.1) induces a well-defined (smooth) section of π via the first equation in (9.1). Similarly, suppose {(Uα , h′α )}α∈A is a collection of trivializations of a rank k′ vector bundle π ′ : V ′ −→ M covering M . A (smooth) vector-bundle homomorphism f˜ : V −→ V ′ covering idM as in (8.5) determines a collection of (smooth) maps ˜ {f˜α : Uα −→ Matk′ ×k R}α∈A s.t. h′α ◦ f˜◦h−1 ∀ (x, v) ∈ Uα ×Rk (9.2) α (x, v) = x, fα (x)v ′ =⇒ f˜α (x)gαβ (x) = g (x)f˜β (x) x ∈ Uα ∩Uβ , α, β ∈ A, (9.3) αβ ′ } ′ ′ where {gαβ α,β∈A is the transition data for the collection of trivializations {hα }α∈A of V . Conversely, a collection of (smooth) maps as in (9.2) satisfying (9.3) induces a well-defined (smooth) vector-bundle homomorphism f˜: V −→ V ′ covering idM as in (8.5) via the equation in (9.2). By the above, starting with a real rank k vector bundle π : V −→ M , we can obtain an open cover {Uα }α∈A of M and a collection of smooth transition maps gαβ : Uα ∩Uβ −→ GLk R α,β∈A . These transition maps satisfy: (VBT1) gαα ≡ Ik , since hαα ≡ hα ◦h−1 α = id; (VBT2) gαβ gβα ≡ Ik , since hαβ ◦hβα = id; (VBT3) gαβ gβγ gγα ≡ Ik , since hαβ ◦hβγ ◦hγα = id. The last condition is known as the (Čech) cocycle condition (more details in Chapter 5 of Warner). It is sometimes written as gα1 α2 gα−1 g ≡ Ik 0 α2 α0 α1 ∀ α0 , α1 , α2 ∈ A. In light of (VBT2), the two versions of the cocycle condition are equivalent. Conversely, given an open cover {Uα }α∈A of M and a collection of smooth maps gαβ : Uα ∩Uβ −→ GLk R α,β∈A that satisfy (VBT1)-(VBT3), we can assemble a rank k vector bundle π ′ : V ′ −→ M as follows. Let G . k ′ α×Uα ×R ∼, where V = α∈A (β, x, v) ∼ α, x, gαβ (x)v ∀ α, β ∈ A, x ∈ Uα ∩Uβ , v ∈ Rk . The relation ∼ is reflexive by (VBT1), symmetric by (VBT2), and transitive by (VBT3) and (VBT2). Thus, ∼ is an equivalence relation, and V ′ carries the quotient topology. Let G q: α×Uα ×Rk −→ V ′ and π ′ : V ′ −→ M, [α, x, v] −→ x, α∈A 47 be the quotient map and the natural projection map (which is well-defined). If β ∈ A and W is a subset of Uβ ×Rk , then G q −1 q(β ×W ) = α×hαβ (W ), where α∈A hαβ : Uα ∩Uβ × R −→ Uα ∩Uβ × Rk , hαβ (x, v) = x, gαβ (x)v . k , then q −1 q(β ×W ) is an open subset of In particular, if β ×W is an open subset of β ×U ×R β F k α∈A α×Uα ×R . Thus, q is an open continuous map. Since its restriction k qα ≡ q|α×Uα ×Rk is injective, (qα (α×Uα ×Rk ), qα−1 ) is a smooth chart on V ′ in the sense of Lemma 2.6. The overlap maps between these charts are the maps hαβ and thus smooth.1 Thus, by Lemma 2.6, these charts induce a smooth structure on V ′ . The projection map π ′ : V ′ −→ M is smooth with respect to this smooth structure, since it induces projection maps on the charts. Since π1 = π ′ ◦ qα : α×Uα ×Rk −→ Uα ⊂ M, the diffeomorphism qα induces a vector-space structure in Vx′ for each x ∈ Uα such that the restriction of qα to each fiber is a vector-space isomorphism. Since the restriction of the overlap map hαβ to x×Rk , with x ∈ Uα ∩Uβ , is a vector-space isomorphism, the vector space structures defined on Vx′ via the maps qα and qβ are the same. We conclude that π ′ : V ′ −→ M is a real vector bundle of rank k. If {Uα }α∈A and gαβ : Uα ∩Uβ −→ GLk R α,β∈A are transition data arising from a vector bundle π : V −→ M , then the vector bundle V ′ constructed in the previous paragraph is isomorphic to V . Let {(Uα , hα )} be the trivializations as above, giving rise to the transition functions gαβ . We define f˜: V −→ V ′ by f˜(v) = α, hα (v) if π(v) ∈ Uα . If π(v) ∈ Uα ∩Uβ , then β, hβ (v) = α, hαβ (hβ (v)) = α, hα (v) ∈ V ′ , i.e. the map f˜ is well-defined (depends only on v and not on α). It is immediate that π ′ ◦ f˜ = π. Since the map k k qα−1 ◦ f˜ ◦ h−1 α : Uα ×R −→ α×Uα ×R is the identity (and thus smooth), f˜ is a smooth map. Since the restrictions of qα and hα to every fiber are vector-space isomorphisms, it follows that so is f˜. We conclude that f˜ is a vector-bundle isomorphism. In summary, a real rank k vector bundle over M determines a set of transition data with values in GLk R satisfying (VBT1)-(VBT3) above (many such sets, of course) and a set of transition data satisfying (VBT1)-(VBT3) determines a real rank-k vector bundle over M . These two processes are well-defined and are inverses of each other when applied to the set of equivalence classes of vector bundles and the set of equivalence classes of transition data satisfying (VBT1)-(VBT3). 1 Formally, the overlap map is (β −→ α)×hαβ . 48 Two vector bundles over M are defined to be equivalent if they are isomorphic as vector bundles over M . Two sets of transition data ′ and gαβ α,β∈A , gαβ α,β∈A with A consisting of all sufficiently small open subsets of M , are said to be equivalent if there exists a collection of smooth functions {fα : Uα −→ GLk R}α∈A such that ′ gαβ = fα gαβ fβ−1 , ∀α, β ∈ A, 2 i.e. the two sets of transition data differ by the action of a Čech 0-chain (more in Chapter 5 of Warner). Along with the cocycle condition on the gluing data, this means that isomorphism classes of real rank k vector bundles over M can be identified with Ȟ 1 (M ; GLk R), the quotient of the space of Čech cocycles of degree one by the subspace of Čech boundaries. Remark 9.2. In Chapter 5 of Warner, Čech cohomology groups, Ȟ m , are defined for (sheafs of) abelian groups. However, the first two groups, Ȟ 0 and Ȟ 1 , generalize to non-abelian groups as well. If π : V −→ M is a complex rank k vector bundle over M , we can similarly obtain transition data for V consisting of an open cover {Uα }α∈A of M and a collection of smooth maps gαβ : Uα ∩Uβ −→ GLk C α,β∈A that satisfies (VBT1)-(VBT3). Conversely, given such transition data, we can construct a complex rank k vector bundle over M . The set of isomorphism classes of complex rank k vector bundles over M can be identified with Ȟ 1 (M ; GLk C). 10 Operations on Vector Bundles Vector bundles can be restricted to smooth submanifolds and pulled back by smooth maps. All natural operations on vector spaces, such as taking quotient vector space, dual vector space, direct sum of vector spaces, tensor product of vector spaces, and exterior powers also carry over to vector bundles via transition functions. Restrictions and pullbacks If N is a smooth manifold, M ⊂ N is an embedded submanifold, and π : V −→ N is a vector bundle of rank k (real or complex) over N , then its restriction to M , π : V |M ≡ π −1 (M ) −→ M, is a vector bundle of rank k over M . It inherits a smooth structure from V by the Slice Lemma (Proposition 5.3) or the Implicit Function Theorem for Manifolds (Theorem 6.3). If {(Uα , hα )} is a collection of trivializations for V −→ N , then {(M ∩Uα , hα |π−1 (M ∩Uα ) )} is a collection of trivializations for V |M −→ M . Similarly, if {gαβ } is transition data for V −→ N , then {gαβ |M ∩Uα ∩Uβ } is 2 According to the discussion around (9.3), such a collection {fα }α∈A corresponds, via trivializations, to an ′ isomorphism between the vector bundles determined by {gαβ }α,β∈A and {gαβ }α,β∈A . 49 transition data for V |M −→ M . If f : M −→ N is a smooth map and π : V −→ N is a vector bundle of rank k, there is a pullback bundle over M : π1 f ∗ V ≡ M ×N V ≡ (p, v) ∈ M ×V : f (p) = π(v) −→ M. (10.1) Note that f ∗ V is the maximal subspace of M ×V so that the diagram f ∗V π2 // V π1 π M f // N commutes. By the Implicit Function Theorem for Maps (Corollary 6.7), f ∗ V is a smooth submanifold of M ×V . By construction, the fiber of π1 over p ∈ M is p×Vf (p) ⊂ M ×V , i.e. the fiber of π over f (p) ∈ N : f ∗ V )p = p × Vf (p) ∀ p ∈ M. (10.2) If {(Uα , hα )} is a collection of trivializations for V −→ N , then {(f −1 (Uα ), hα ◦f )} is a collection of trivializations for f ∗ V −→ M . Similarly, if {gαβ } is transition data for V −→ N , then {gαβ ◦ f } is transition data for f ∗ V −→ M . The case discussed in the previous paragraph corresponds to f being the inclusion map. Lemma 10.1. If f˜: V −→ V ′ is a vector-bundle homomorphism covering a smooth map f : M −→ N as in (8.4), there exists a bundle homomorphism φ : V −→ f ∗ V ′ so that the diagram f˜ [email protected] @ φ @@ @@ @@ π M // f ∗ V ′ yy yy y y π ||yy 1 f π2 %% // V ′ π′ // N commutes. The map φ is defined by φ(v) = π(v), f˜(v) . φ : V −→ M ×V ′ , Since f ◦ π = π ′ ◦ f˜, φ(v) ∈ f ∗ V ′ ≡ M ×N V ′ ≡ (p, v ′ ) ∈ M ×V ′ : f (p) = π ′ (v ′ ) . Since f ∗ V ′ ⊂ M × V ′ is a smooth embedded submanifold, the map φ : V −→ f ∗ V ′ obtained by restricting the range is smooth; see Proposition 5.5. The above diagram commutes by the construction of φ. Since f˜ is linear on each fiber of V , so is φ. If f : M −→ N is a smooth map, then dp f : Tp M −→ Tf (p) N is a linear map which varies smoothly with p. It thus gives rises to a smooth map, df : T M −→ T N, v −→ dπ(v) f (v). 50 (10.3) However, this description of df gives no indication that df maps v ∈ Tp M to Tf (p) N or that this map is linear on each Tp M . One way to fix this defect is to state that (10.3) is a bundle homomorphism covering the map f : M −→ N , i.e. that the diagram TM df // T N π (10.4) π′ f M // N commutes. By Lemma 10.1, df then induces a vector-bundle homomorphism from T M to f ∗ T N so that the diagram T MD DD DD D π DD !! df // f ∗ T N _π_2 _// T N xx ′ x x xx π1 π x {{x f M _ _ _ _ _ _ _ _ _// N (10.5) commutes. The triangular part of (10.5) is generally the preferred way of describing df . The description (10.4) factors through the triangular part of (10.5), as indicated by the dashed arrows. The triangular part of (10.5) also leads to a more precise statement of the Implicit Function Theorem, which is rather useful in topology of manifolds; see Theorem 10.11 below. If π : V −→ N is a smooth vector bundle, f : M −→ N is a smooth map, and s : N −→ V is a bundle section of V , then f ∗ s : M −→ f ∗ V, {f ∗ s}(p) = p, s(f (p)) ∈ f ∗ V ≡ M ×N V ⊂ M ×V, is a bundle section of f ∗ V −→ M . If s is smooth, then f ∗ s : M −→ M ×V is a smooth map with the image in M ×N V . Since M ×N V ⊂ M ×V is an embedded submanifold, f ∗ s : M −→ f ∗ V is a smooth map by Proposition 5.5. Thus, a smooth map f : M −→ N induces a homomorphism of vector spaces f ∗ : Γ(N ; V ) −→ Γ(M ; f ∗ V ), s −→ f ∗ s, (10.6) which is also a homomorphism of modules with respect to the ring homomorphism f ∗ : C ∞ (N ) −→ C ∞ (M ), g −→ g ◦ f . In the case of tangent bundles, the homomorphism (10.6) is compatible with the Lie algebra structures on the spaces of vector fields, as described by the following lemma. Lemma 10.2. Let f : M −→ N be a smooth map. If X1 , X2 ∈ VF(M ) and Y1 , Y2 ∈ VF(N ) are smooth vector fields on M and N , respectively, such that df (Xi ) = f ∗ Yi ∈ Γ(M ; f ∗ T N ) for i = 1, 2, then df [X1 , X2 ] = f ∗ [Y1 , Y2 ]. This is checked directly from the relevant definitions. The pullback operation on vector bundles also extends to homomorphisms. Let f : M −→ N be a smooth map and πV : V −→ N and πW : W −→ N be vector bundles. Any vector-bundle 51 homomorphism ϕ : V −→ W over N induces a vector-bundle homomorphism f ∗ ϕ : f ∗ V −→ f ∗ W over M so that the diagram f ∗ V2 QQ 22 QQQQQf ∗ ϕ QQ 2 22 22 π1 22 22 22 2 M π2 QQQ QQQ (( f ∗W y π1 yy y y y || y y f // V O // OOO // OOOOϕ OOO πV / OOO // O'' π2 // W // } // } }} // / ~~}}}}πW // N (10.7) commutes. The vector-bundle homomorphism f ∗ ϕ is given by (f ∗ ϕ)p = id×ϕf (p) : (f ∗ V )p = p×Vf (p) −→ (f ∗ W )p = p×Wf (p) , (p, v) −→ p, ϕp (v) , where ϕp is the restriction of ϕ to the fiber Vf (p) = πV−1 (f (p)) over f (p) ∈ N . Subbundles and foliations by submanifolds Definition 10.3. Let M be a smooth manifold. (1) A rank k′ subbundle of a vector bundle π : V −→ M is a smooth submanifold V ′ of V such that π|V ′ : V ′ −→ M is a vector bundle of rank k′ . (2) A rank k distribution on M is a rank k subbundle of T M −→ M . A subbundle of course cannot have a larger rank than the ambient bundle; so rk V ′ ≤ rk V in Definition 10.3 and the equality holds if and only if V ′ = V . By Exercise 17, the requirement that π|V ′ : V ′ −→ M is a vector bundle of rank k′ can be replaced by the condition that Vp′ ≡ Vp ∩V ′ is a k′ -dimensional linear subspace of Vp for all p ∈ M . If f : M −→ N is an immersion, the bundle homomorphism df as in (10.5) is injective and the image of df in f ∗ T N is a subbundle of f ∗ T N . In the case M ⊂ N is an embedded submanifold and f is the inclusion map, we identify T M with the image of dι in f ∗ T N = T N |M . By Lemma 10.2, if Y1 , Y2 ∈ VF(N ) are smooth vector fields on N , then Y1 M , Y2 M ∈ VF(M ) ⊂ Γ(M ; T N |M ) =⇒ [Y1 , Y2 ]M ∈ VF(M ) ⊂ Γ(M ; T N |M ). Definition 10.4. Let N be a smooth manifold. (1) A collection {ια : Mα −→ N }α∈A of injective immersions from m-manifolds is a foliation of N n if the collection {Im ια }α∈A covers N and for every q ∈ N there exists a smooth chart ψ : V −→ Rm×Rn−m around q such that the image under ια of every connected subset U ⊂ ι−1 α (V ) under ψ is contained in ψ −1 (Rm ×y) for some y ∈ Rn−m (dependent on U ). (2) A foliation {ια : Mα −→ N }α∈A of N is proper if ια is an embedding and the images of ια partition N (their union covers M and any two of them are either disjoint or the same). Thus, a foliation of N consists of regular immersions that cover N and are regular in a systematic way (all of them correspond to horizontal slices in a single coordinate chart); see Figure 2.3. Since −1 manifolds are second-countable and the subset ι−1 α (V ) ⊂ Mα in Definition 10.4 is open, ια (ια (V )) 52 Rn−m ι1 (M1 ) ∩ V ι2 (M2 ) ∩ V Rm Figure 2.3: A foliation of N in a smooth chart V . is contained in at most countably many of the horizontal slices ψ −1 (Rm ×y). The images of dια in T N determine a rank m distribution D on N . By Lemma 10.2, if Y1 , Y2 ∈ VF(N ) are vector fields on N , then Y1 , Y2 ∈ Γ(N ; D) ⊂ VF(N ) =⇒ [Y1 , Y2 ] ∈ Γ(N ; D) ⊂ VF(N ). (10.8) Definition 10.5. Let D ⊂ T N be a distribution on a smooth manifold N . An injective immersion ι : M −→ N is integral for D if Im dp ι = Dι(p) ⊂ Tι(p) N ∀ p ∈ M. If ι : M −→ N is an integrable injective immersion for a distribution D on N , then in particular dim M = rk D. If N admits a foliation {ια : Mα −→ N }α∈A by injective immersions integral to a distribution D on N , then Γ(N ; D) ⊂ VF(N ) is a Lie subalgebra. By Frobenius Theorem, the converse is also true. Example 10.6. The collection of embeddings ια : Rm −→ Rn = Rm ×Rn−m , α ∈ Rn−m , ια (x) = (x, α), is a proper foliation of Rn by m-manifolds. The corresponding distribution D ⊂ T Rn is described by D = Rn × (Rm ×0) ⊂ Rn × Rn = T Rn . Example 10.7. The collection of embeddings ια : S 1 −→ S 2n+1 ⊂ Cn+1 , ια (eiθ ) = eiθ α, α ∈ S 2n+1 , is a proper foliation of S 2n+1 by circles. The corresponding distribution D ⊂ T S 2n+1 is described by D = (p, irp) : p ∈ S 2n+1 , r ∈ R ⊂ T S 2n+1 ⊂ T Cn+1 S 2n+1 = S 2n+1 ×Cn+1 . The embedded submanifolds of this foliations are the fibers of the quotient projection map π : S 2n+1 −→ S 2n+1 /S 1 = CP n of Example 1.10. This is an S 1 -bundle over CP n . In general, the fibers of the projection map π : N −→ B of any smooth fiber bundle form a proper foliation of the total space N of the bundle. The corresponding distribution D ⊂ T N is then the vertical tangent bundle of π: Dp = ker dp π ⊂ Tp N 53 ∀ p ∈ N. Example 10.8. An example of a foliation, which is not proper, is provided by the skew lines on the torus of the same irrational slope η: ια : R −→ S 1 ×S 1 , ια (s) = αeis , eiηs , α ∈ S 1 ⊂ C. If η ∈ Q, this foliation is proper. In either case, the corresponding distribution D on S 1 × S 1 is described by D(eit1 ,eit2 ) = d(t1 ,t2 ) q {(r, ηr) ∈ R2 = T(t1 ,t2 ) R2 : r ∈ R} , where q : R2 −→ S 1 ×S 1 the usual covering map. Quotient and normal bundles If V is a vector space (over R or C) and V ′ ⊂ V is a linear subspace, then we can form the quotient vector space, V /V ′ . If W is another vector space, W ′ ⊂ W is a linear subspace, and g : V −→ W is a linear map such that g(V ′ ) ⊂ W ′ , then g descends to a linear map between the quotient spaces: ḡ : V /V ′ −→ W/W ′ . If we choose bases for V and W such that the first few vectors in each basis form bases for V ′ and W ′ , then the matrix for g with respect to these bases is of the form: A B g= . 0 D The matrix for ḡ is then D. If g is an isomorphism from V to W that restricts to an isomorphism from V ′ to W ′ , then ḡ is an isomorphism from V /V ′ to W/W ′ . Any vector-space homomorphism ϕ : V −→ W such that V ′ ⊂ ker ϕ descends to a homomorphism ϕ̄ so that the diagram ϕ V q y y y ϕ̄ // W y<< V /V ′ commutes. If V ′ ⊂ V is a subbundle, we can form a quotient bundle, V /V ′ −→ M , such that (V /V ′ )p = Vp /Vp′ ∀ p ∈ M. The topology on V /V ′ is the quotient topology for the natural surjective map q : V −→ V /V ′ . The vector-bundle structure on V /V ′ is determined from those of V and V ′ by requiring that q be a smooth vector-bundle homomorphism. Thus, if s is a smooth section of V , then q ◦s is a smooth section of V /V ′ ; so, there is a homomorphism Γ(M ; V ) −→ Γ(M ; V /V ′ ), s −→ q ◦ s, of C ∞ (M )-modules. There is also a short exact sequence3 of vector bundles over M , q 0 −→ V ′ −→ V −→ V /V ′ −→ 0, 3 exact means that at each position the kernel of the outgoing vector-bundle homomorphism equals the image of the incoming one; short means that it consists of five terms with zeros (rank 0 vector bundles) at the ends 54 where the zeros denote the zero vector bundle M×0 −→ M . We can choose a system of trivializations {(Uα , hα )}α∈A of V such that ′ ∀ α ∈ A. (10.9) hα V ′ |Uα = Uα × (Rk ×0) ⊂ Uα ×Rk ′ Let qk′ : Rk −→ Rk−k be the projection onto the last (k −k′ ) coordinates. The trivializations for V /V ′ are then given by {(Uα , {id × qk′ } ◦ hα )}. Alternatively, if {gαβ } is transition data for V such that the upper-left k′ ×k′ -submatrices of gαβ correspond to V ′ (as is the case for the above trivializations hα ) and ḡαβ is the lower-right (k−k′ )×(k−k′ ) matrix of gαβ , then {ḡαβ } is transition data for V /V ′ . Any vector-bundle homomorphism ϕ : V −→ W over M such that ϕ(v) = 0 for all v ∈ V ′ descends to a vector-bundle homomorphism ϕ̄ so that ϕ = ϕ̄◦q. We leave proofs of the following lemmas as an exercise. Lemma 10.9. If f : M −→ N is a smooth map and W, W ′ −→ N are smooth vector bundles, f ∗ W/W ′ ) ≈ (f ∗ W )/(f ∗ W ′ ) as vector bundles over M . Lemma 10.10. Let V −→ M and W −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. A vector-bundle homomorphism f˜: V −→ W covering f as in (8.4) and vanishing on a subbundle V ′ ⊂ V induces a vector-bundle homomorphism f¯: V /V ′ −→ W covering f ; this induced homomorphism is smooth if the homomorphism f˜ is smooth. If ι : Y −→ N is an immersion, the image of dι in ι∗ T N is a subbundle of ι∗ T N . In this case, the quotient bundle, NN ι ≡ ι∗ T N Im dι −→ Y, is called the normal bundle for the immersion ι. If Y is an embedded submanifold and ι is the inclusion map, T Y is a subbundle of ι∗ T N = T N |Y and the quotient subbundle, NN Y ≡ NN ι = ι∗ T N Im dι = T N |Y T Y −→ Y, is called the normal bundle of Y in N ; its rank is the codimension of Y in N . If f : M −→ N is a smooth map and X ⊂ M is an embedded submanifold, the vector-bundle homomorphism df in (10.5) restricts (pulls back by the inclusion map) to a vector-bundle homomorphism df |X : T M |X −→ (f ∗ T N )X over X, which can be composed with the inclusion homomorphism T X −→ T M |X , df |X T X −→ T M |X −→ (f ∗ T N )X . If in addition f (X) ⊂ Y , then the above sequence can be composed with the f ∗ -pullback of the projection homomorphism q : T N |Y −→ NN Y , f ∗q df |X T X −→ T M |X −→ (f ∗ T N )X −→ f ∗ NN Y . 55 (10.10) This composite vector-bundle homomorphism is 0, since dx f (v) ∈ Tf (x) Y for all x ∈ X. Thus, it descends to a vector-bundle homomorphism df : NM X −→ f ∗ NN Y (10.11) over X. If f ⊤ ∩ N Y as in (6.1), then the map T M |X −→ f ∗ NN Y in (10.10) is onto and thus the vector-bundle homomorphism (10.11) is surjective on every fiber. Finally, if X = f −1 (Y ), the ranks of the two bundles in (10.11) are the same by the last statement in Theorem 6.3, and so (10.11) is an isomorphism of vector bundles over X. Combining this observation with Theorem 6.3, we obtain a more precise statement of the latter. Theorem 10.11. Let f : M −→ N be a smooth map and Y ⊂ N an embedded submanifold. If f⊤ ∩ N Y as in (6.1), then X ≡ f −1 (Y ) is an embedded submanifold of M and the differential df induces a vector-bundle isomorphism df NM XF FF FF F π FFF "" M // f ∗ (NN Y ) tt tt t tt π1 t zz t (10.12) Since the ranks of NM X and f ∗ (NN Y ) are the codimensions of X in M and Y in N , respectively, this theorem implies Theorem 6.3. If Y = {q} for some q ∈ N , then NN Y is a trivial vector bundle and thus so is NM X ≈ f ∗ (NN Y ). For example, the unit sphere S m ⊂ Rm+1 has trivial normal bundle, because S m = f −1 (1), where f : Rm+1 −→ R, f (x) = |x|2 . A trivialization of the normal bundle to S m is given by T Rm+1 /T S m −→ S m ×R, (x, v) −→ (x, x·v). Corollary 10.12. Let f : X −→ M and g : Y −→ M be smooth maps. If f⊤ ∩M g as in (6.5), then the space X ×M Y ≡ (x, y) ∈ X ×Y : f (x) = g(y) is an embedded submanifold of X ×Y and the differential df induces a vector-bundle isomorphism NX×Y (X ×M Y ) d(f ◦πX )+d(g◦πY ) PPP PPP PP π PPPP (( X ×M Y // π ∗ f ∗ T M = π ∗ g ∗ T M X Y lll l l ll lll uu ll l (10.13) Furthermore, the projection map π1 = πX : X ×M Y −→ X is injective (immersion) if g : Y −→ M is injective (immersion). This corollary is obtained by applying Theorem 10.11 to the smooth map f ×g : X ×Y −→ M ×M. All other versions of the Implicit Function Theorem stated in these notes are special cases of this corollary. 56 Direct sums If V and V ′ are two vector spaces, we can form a new vector space, V ⊕V ′ = V ×V ′ , the direct sum of V and V ′ . There are natural inclusions V, V ′ −→ V ⊕V ′ and projections V ⊕V ′ −→ V, V ′ . Linear maps f : V −→ W and f ′ : V ′ −→ W ′ induce a linear map f ⊕f ′ : V ⊕V ′ −→ W ⊕W ′ . If we choose bases for V , V ′ , W , and W ′ so that f and f ′ correspond to some matrices A and D, then with respect to the induced bases for V ⊕V ′ and W ⊕W ′ , A 0 f ⊕g = . 0 D If π : V −→ M and π ′ : V ′ −→ M are smooth vector bundles, we can form their direct sum, V ⊕V ′ , so that ∀ p ∈ M. (V ⊕V )p = Vp ⊕Vp′ The vector-bundle structure on V ⊕ V ′ is determined from those of V and V ′ by requiring that either the natural inclusion maps V, V ′ −→ V ⊕V ′ or the projections V ⊕V ′ −→ V, V ′ be smooth vector-bundle homomorphisms over M . Thus, if s and s′ are sections of V and V ′ , then s⊕s′ is a smooth section of V ⊕V ′ if and only if s and s′ are smooth. So, the map Γ(M ; V ) ⊕ Γ(M ; V ′ ) −→ Γ(M ; V ⊕V ′ ), (s, s′ ) −→ s⊕s′ , s⊕s′ (p) = s(p) ⊕ s′ (p) ∀ p ∈ M, ′ } are transition data for V and V ′ , is an isomorphism of C ∞ (M )-modules. If {gαβ } and {gαβ ′ }, i.e. we put the first matrix in the top left corner transition data for V ⊕V ′ is given by {gαβ ⊕gαβ and the second matrix in the bottom right corner. Alternatively, π×π ′ : V ×V ′ −→ M ×M is a smooth vector bundle with respect to the product structures and V ⊕ V ′ = d∗ (V ×V ′ ), where d : M −→ M ×M, (10.14) d(p) = (p, p) is the diagonal embedding. The operation ⊕ is easily seen to be commutative and associative (the resulting vector bundles are isomorphic). If τ0 = M −→ M is trivial rank 0 bundle, τ0 ⊕ V ≈ V for every vector bundle V −→ M . If n ∈ Z≥0 , let nV = V . . ⊕ V} ; | ⊕ .{z n by convention; 0V = τ0 . We leave proofs of the following lemmas as an exercise. 57 Lemma 10.13. If f : M −→ N is a smooth map and W, W ′ −→ N are smooth vector bundles, f ∗ W ⊕ W ′ ) ≈ (f ∗ W ) ⊕ (f ∗ W ′ ) as vector bundles over M . Lemma 10.14. Let V, V ′ −→ M and W, W ′ −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. Vector-bundle homomorphisms f˜: V −→ W and f˜′ : V ′ −→ W ′ covering f as in (8.4) induce a vector-bundle homomorphism f˜⊕ f˜′ : V ⊕V ′ −→ W ⊕W ′ covering f ; this induced homomorphism is smooth if and only if f˜ and f˜′ are smooth. If V, V ′ −→ M are vector bundles, then V and V ′ are vector subbundles of V ⊕V ′ . It is immediate that V ⊕V ′ /V = V ′ and V ⊕V ′ /V ′ = V. These equalities hold in the holomorphic category as well (i.e. when the bundles and the base manifold carry complex structures and all trivializations and transition maps are holomorphic). Conversely, if V ′ is a subbundle of V , by Section 11 below V ≈ (V /V ′ ) ⊕ V ′ as smooth vector bundles, real or complex. However, if V and V ′ are holomorphic bundles, V may not have the same holomorphic structure as (V /V ′ )⊕ V ′ (i.e. the two bundles are isomorphic as smooth vector bundles, but not as holomorphic ones). Dual bundles If V is a vector space (over R or C), the dual vector space is the space of the linear homomorphisms to the field (R or C, respectively): V ∗ = HomR V, R) or V ∗ = HomC V, C). A linear map g : V −→ W between two vector spaces induces a dual map in the “opposite” direction: ∗ g∗ : W ∗ −→ V ∗ , g (L) (v) = L g(v) ∀ L ∈ W ∗ , v ∈ V. If V = Rk and W = Rn , then g is given by an n×k-matrix, and its dual is given by the transposed k×n-matrix. If π : V −→ M is a smooth vector bundle of rank k (say, over R), the dual bundle of V is a vector bundle V ∗ −→ M such that (V ∗ )p = Vp∗ ∀ p ∈ M. The vector-bundle structure on V ∗ is determined from that of V by requiring that the natural map V ⊕V ∗ = V × M V ∗ −→ R (or C), 58 (v, L) −→ L(v), (10.15) be smooth. Thus, if s and ψ are smooth sections of V and V ∗ , ψ(s) : M −→ R, {ψ(s)}(p) = ψ(p) s(p) , is a smooth function on M . So, the map Γ(M ; V ) × Γ(M ; V ∗ ) −→ C ∞ (M ), (s, ψ) −→ ψ(s), is a nondegenerate pairing of C ∞ (M )-modules. If {gαβ } is transition data for V , i.e. the transitions between smooth trivializations are given by k k hα ◦h−1 (p, v) −→ p, gαβ (p)v , β : Uα ∩Uβ × R −→ Uα ∩Uβ × R , the dual transition maps are then given by Uα ∩Uβ × Rk −→ Uα ∩Uβ × Rk , (p, v) −→ p, gαβ (p)tr v . However, these maps reverse the direction, i.e. they go from the α-side to the β-side. To fix this problem, we simply take the inverse of gαβ (p)tr : Uα ∩Uβ × Rk −→ Uα ∩Uβ × Rk , (p, v) −→ p, {gαβ (p)tr }−1 v . tr )−1 }. As an example, if V is a line bundle, then g So, transition data for V ∗ is {(gαβ αβ is a smooth nowhere-zero function on Uα ∩ Uβ and (g ∗ )αβ is the smooth function given by 1/gαβ . We leave proofs of the following lemmas as an exercise. Lemma 10.15. If f : M −→ N is a smooth map and W −→ N is a smooth vector bundle, f ∗ (W ∗ ) ≈ (f ∗ W )∗ as vector bundles over M . Lemma 10.16. Let V −→ M and W −→ N be vector bundles over smooth manifolds and f : M −→ N a diffeomorphism. A vector-bundle homomorphism f˜ : V −→ W covering f as in (8.4) induces a vector-bundle homomorphism f˜∗ : W ∗ −→ V ∗ covering f −1 ; this induced homomorphism is smooth if and only if the homomorphism f˜ is. The cotangent bundle of a smooth manifold M , π : T ∗ M −→ M , is the dual of its tangent bundle, T M −→ M , i.e. T ∗ M = (T M )∗ . For each p ∈ M , the fiber of the cotangent bundle over p is the cotangent space Tp∗ M of M at p; see Definition 3.7. A section α : M −→ T ∗ M of T ∗ M is called a 1-form on M ; it assigns to each p ∈ M a linear map αp ≡ α(p) : Tp M −→ R. If in addition X is a vector field, then α(X) : M −→ R, α(X) (p) = αp X(p) , is a function on M . The section α is smooth if and only if α(X) ∈ C ∞ (M ) for every smooth vector field X on M . If ϕ = (x1 , . . . , xm ) : U −→ Rm is a smooth chart, the sections ∂ ∂ ,... , ∈ VF(U ) ∂x1 ∂xm 59 form a basis for VF(U ) as a C ∞ (U )-module. Since ∂ dp xi = δij ∀ i, j = 1, 2, . . . , m, ∂xj dxi (X) ∈ C ∞ (U ) for all X ∈ VF(U ) and {dp xi }i is a basis for Tp∗ M for all p ∈ U . Thus, dxi is a smooth section of T ∗ M over U and the inverse of the map U ×Rm −→ T ∗ M |U , (p, c1 , . . . , cm ) −→ c1 dp x1 + . . . + cm dp xm , is a trivialization of T ∗ M over U ; see Section 8. By (4.16), this inverse is given by ∂ ∂ ∗ m T M |U −→ U ×R , u −→ π(u), u ,...,u , ∂x1 ∂xm where π : T ∗ M −→ M is the projection map. Thus, a 1-form α on M is smooth if and only if for every smooth chart ϕα = (x1 , . . . , xm ) : Uα −→ Rm the coefficient functions ∂ ∂ , . . . , cm = : U −→ R, αp ≡ c1 (p)dp x1 + . . . + cm (p)dp xm ∀ p ∈ U, c1 = α ∂x1 ∂xm are smooth. The C ∞ (M )-module of 1-forms on M is denoted by E 1 (M ). Tensor products If V and V ′ are two vector spaces, we can form a new vector space, V ⊗V ′ , the tensor product of V and V ′ . If g : V −→ W and g ′ : V ′ −→ W ′ are linear maps, they induce a linear map g⊗g ′ : V ⊗V ′ −→ W ⊗W ′ . ′ } for V , V ′ , W , and W ′ , respectively, then {e ⊗e′ } If we choose bases {ej }, {e′n }, {fi }, and {fm j n (j,n) ′ ′ and {fi ⊗fm }(i,m) are bases for V ⊗V and W ⊗W ′ . If the matrices for g and g′ with respect to the ′ ′ ) chosen bases for V , V ′ , W , and W ′ are (gij )i,j and (gmn m,n , then the matrix for g⊗g with respect ′ ′ ′ to the induced bases for V ⊗V and W⊗W is (gij gmn )(i,m),(j,n) . The rows of this matrix are indexed by the pairs (i, m) and the columns by the pairs (j, n). In order to actually write down the matrix, we need to order all pairs (i, m) and (j, n). If the vector spaces V and W are one-dimensional, g corresponds to a single number gij , while g⊗g ′ corresponds to the matrix (gmn )m,n multiplied by this number. If π : V −→ M and π ′ : V ′ −→ M are smooth vector bundles, we can form their tensor product, V ⊗V ′ , so that (V ⊗V ′ )p = Vp ⊗Vp′ ∀ p ∈ M. The topology and smooth structure on V ⊗V ′ are determined from those of V and V ′ by requiring that if s and s′ are smooth sections of V and V ′ , then s ⊗ s′ is a smooth section of V ⊗V ′ . So, the map Γ(M ; V ) ⊗ Γ(M ; V ′ ) −→ Γ(M ; V ⊗V ′ ), (s, s′ ) −→ s⊗s′ , s⊗s′ (p) = s(p) ⊗ s′ (p) ∀ p ∈ M, 60 ′ } are transiis a homomorphism of C ∞ (M )-modules (but not an isomorphism). If {gαβ } and {gαβ ′ ′ ′ tion data for V and V , then transition data for V ⊗V is given by {gαβ ⊗gαβ }, i.e. we construct a ′ ′ } as in the previous paragraph. If V and V ′ matrix-valued function gαβ ⊗gαβ from {gαβ } and {gαβ ′ are line bundles, then gαβ and gαβ are smooth nowhere-zero functions on Uα ∩Uβ and (g⊗g ′ )αβ is ′ . the smooth function given by gαβ gαβ The operation ⊗ is easily seen to be commutative and associative (the resulting vector bundles are isomorphic). If τ1 −→ M is the trivial line bundle, τ1 ⊗ V ≈ V for every vector bundle V −→ M is a vector bundle. If n ∈ Z+ , let V ⊗(−n) = (V ∗ )⊗n ≡ |V ∗ ⊗ .{z . . ⊗ V }∗ ; V ⊗n = V . . ⊗ V} , | ⊗ .{z n n by convention, V ⊗0 = τ1 . We leave proofs of the following lemmas as an exercise. Lemma 10.17. If f : M −→ N is a smooth map and W, W ′ −→ N are smooth vector bundles, f ∗ W ⊗ W ′ ) ≈ (f ∗ W ) ⊗ (f ∗ W ′ ) as vector bundles over M . Lemma 10.18. Let V, V ′ −→ M and W, W ′ −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. Vector-bundle homomorphisms f˜: V −→ W and f˜: V ′ −→ W ′ covering f as in (8.4) induce a vector-bundle homomorphism f˜⊗ f˜′ : V ⊗V ′ −→ W ⊗W ′ covering f ; this induced homomorphism is smooth if f˜ and f˜′ are smooth. Lemma 10.19. Let V, V ′ −→ M and W −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. A bundle map f˜: V ⊕V ′ = V × M V −→ W covering f as in (8.4) such that the restriction of f˜ to each fiber Vp×Vp is linear in each component induces a vector-bundle homomorphism f¯: V ⊗V ′ −→ W covering f ; this induced homomorphism is smooth if the homomorphism f˜ is. 61 Exterior products If V is a vector space and k is a nonnegative integer, we can form the k-th exterior power, Λk V , of V . A linear map g : V −→ W induces a linear map Λk g : Λk V −→ Λk W. If n is a nonnegative integer, let Sk (n) be the set of increasing k-tuples of integers between 1 and n: Sk (n) = (i1 , . . . , ik ) ∈ Zk : 1 ≤ i1 < i2 < . . . < ik ≤ n . If {ej }j=1,...,n and {fi }i=1,...,m are bases for V and W , then {eη }η∈Sk (n) and {fµ }µ∈Sk (m) are bases for Λk V and Λk W , where e(η1 ,...,ηk ) = eη1 ∧ . . . ∧ eηk and f(µ1 ,...,µk ) = fµ1 ∧ . . . ∧ fµk . If (gij )i=1,...,m,j=1,...,n is the matrix for g with respect to the chosen bases for V and W , then det (gµr ηs )r,s=1,...,k (µ,η)∈I (m)×I (n) k k is the matrix for Λk g with respect to the induced bases for Λk V and Λk W . The rows and columns of this matrix are indexed by the sets Sk (m) and Sk (n), respectively. The (µ, η)-entry of the matrix is the determinant of the k × k-submatrix of (gij )i,j with the rows and columns indexed by the entries of µ and η, respectively. In order to actually write down the matrix, we need to order the sets Sk (m) and Sk (n). If k = m = n, then Λk V and Λk W are one-dimensional vector spaces, called the top exterior power of V and W , with bases and f1 ∧ . . . ∧ fk . e1 ∧ . . . ∧ ek With respect to these bases, the homomorphism Λk g corresponds to the number det(gij )i,j . If k > n (or k > m), then Λk V (or Λk W ) is the zero vector space and the corresponding matrix is empty. If π : V −→ M is a smooth vector bundle, we can form its k-th exterior power, Λk V, so that (Λk V )p = Λk Vp ∀ p ∈ M. The topology and smooth structure on Λk V are determined from those of Λk V by requiring that if s1 , . . . , sk are smooth sections of V , then s1 ∧. . .∧sk is a smooth section of Λk V . Thus, the map Λk Γ(M ; V ) −→ Γ(M ; Λk V ), (s1 , . . . , sk ) −→ s1 ∧. . .∧sk , {s1 ∧. . .∧sk }(p) = s1 (p)∧. . .∧sk (p) ∀ p ∈ M, is a homomorphism of C ∞ (M )-modules (but not an isomorphism). If {gαβ } is transition data for V , then transition data for Λk V is given by {Λk gαβ }, i.e. we construct a matrix-valued function Λk gαβ from each matrix gαβ as in the previous paragraph. As an example, if the rank of V is k, then the transition data for the line bundle Λk V , called the top exterior power of V , is {det gαβ }. By definition, Λ0 V = τ1R is the trivial line bundle over M . It follows directly from the definitions that if V −→ M is a vector bundle of rank k and L −→ M is a line bundle (vector bundle of rank one), then Λtop (V ⊕L) ≡ Λk+1 (V ⊕L) = Λk V ⊗ L ≡ Λtop V ⊗ L. 62 More generally, if V, W −→ M are any two vector bundles, then Λtop (V ⊕W ) = (Λtop V ) ⊗ (Λtop W ) and Λk (V ⊕W ) = M (Λi V )⊗(Λj W ). i+j=k We leave proofs of the following lemmas as exercises. Lemma 10.20. If f : M −→ N is a smooth map, W −→ N is a smooth vector bundle, and k ∈ Z≥0 , f ∗ Λk W ) ≈ Λk (f ∗ W ) as vector bundles over M . Lemma 10.21. Let V −→ M be a vector bundle. If k, l ∈ Z≥0 , the map Γ(M ; Λk V ) ⊗ Γ(M ; Λl V ) −→ Γ(M ; Λk+l V ) (s1 , s2 ) −→ s1 ∧s2 , {s1 ∧s2 }(p) = s1 (p)∧s2 (p) ∀ p ∈ M, is a well-defined homomorphism of C ∞ (M )-modules. Lemma 10.22. Let V −→ M and W −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. A vector-bundle homomorphism f˜: V −→ W covering f as in (8.4) induces a vectorbundle homomorphism Λk f˜: Λk V −→ Λk W covering f ; this induced homomorphism is smooth if the homomorphism f˜ is. Lemma 10.23. Let V −→ M and W −→ N be vector bundles over smooth manifolds and f : M −→ N a smooth map. A bundle homomorphism f˜: kV ≡ V ×M . . .×M V −→ W {z } | k covering f as in (8.4) such that the restriction of f˜ to each fiber Vpk is linear in each component and alternating induces a vector-bundle homomorphism f¯: Λk V −→ W covering f ; this induced homomorphism is smooth if the homomorphism f˜ is. Remark: For complex vector bundles, the above constructions (exterior power, tensor product, direct sum, etc.) are always done over C, unless specified otherwise. So if V is a complex vector bundle of rank k over M , the top exterior power of V is the complex line bundle Λk V over M (could also be denoted as ΛkC V ). In contrast, if we forget the complex structure of V (so that it becomes a real vector bundle of rank 2k), then its top exterior power is the real line bundle Λ2k V (could also be denoted as Λ2k R V ). If M is a smooth manifold, a section of the bundle Λk (T ∗ M ) −→ M is called a k-form on M . A smooth nowhere-vanishing section s of Λtop (T ∗ M ), i.e. s(p) ∈ Λtop (Tp∗ M ) − 0 ∀ p ∈ M, is called a volume form on M ; Corollary 12.2 below provides necessary and sufficient conditions for such a section to exist. The space of smooth k-forms on M is often denoted by E k (M ), rather than Γ(M ; Λk (T ∗ M )). 63 11 Metrics on Fibers If V is a vector space over R, a positive-definite inner-product on V is a symmetric bilinear map h·, ·i : V ×V −→ R, (v, w) −→ hv, wi, s.t. hv, vi > 0 ∀ v ∈ V −0. If h, i and h, i′ are positive-definite inner-products on V and a, a′ ∈ R̄+ are not both zero, then ah, i+a′ h, i′ : V ×V −→ R, ah, i+a′ h, i′ (v, w) = ahv, wi + a′ hv, wi′ , is also a positive-definite inner-product. If W is a subspace of V and h, i is a positive-definite inner-product on V , let W ⊥ = v ∈ V : hv, wi = 0 ∀ w ∈ W be the orthogonal complement of W in V . In particular, V = W ⊕ W ⊥. Furthermore, the quotient projection map π : V −→ V /W induces an isomorphism from W ⊥ to V /W so that V ≈ W ⊕ (V /W ). If M is a smooth manifold and V −→ M is a smooth real vector bundle of rank k, a Riemannian metric on V is a positive-definite inner-product in each fiber Vx ≈ Rk of V that varies smoothly with x ∈ M . Formally, the smoothness requirement is one of the following equivalent conditions: (a) the map h, i : V × M V −→ R is smooth; (b) the section h, i of the vector bundle (V ⊗V )∗ −→ M is smooth; (c) if s1 , s2 are smooth sections of the vector bundle V −→ M , then the map s1 , s2 : M −→ R, p −→ s1 (p), s2 (p) , is smooth; (d) if h : V |U −→ U ×Rk is a trivialization of V , then the matrix-valued function, B : U −→ Matk R s.t. h−1 (p, v), h−1 (p, w) = v t B(p)w ∀ p ∈ U, v, w ∈ Rk , is smooth. Every real vector bundle admits a Riemannian metric. Such a metric can be constructed by covering M by a locally finite collection of trivializations for V and patching together positivedefinite inner-products on each trivialization using a partition of unity; see Definition 11.1 below. If W is a subspace of V and h, i is a Riemannian metric on V , let W ⊥ = v ∈ V : hv, wi = 0 ∀ w ∈ W 64 be the orthogonal complement of W in V . Then W ⊥ −→ M is a vector subbundle of V and V = W ⊕ W ⊥. Furthermore, the quotient projection map π : V −→ V /W induces a vector bundle isomorphism from W ⊥ to V /W so that V ≈ W ⊕ (V /W ). Definition 11.1. A smooth partition of unity subordinate to the open cover {Uα }α∈A of a smooth manifold M is a collection {ηα }α∈A of smooth functions on M with values in [0, 1] such that (PU1) the collection {supp ηα }α∈A is locally finite; (PU2) supp ηα ⊂ Uα for every α ∈ A; X (PU3) ηα ≡ 1. α∈A If V is a vector space over C, a nondegenerate Hermitian inner-product on V is a map h·, ·i : V ×V −→ C, (v, w) −→ hv, wi, which is C-antilinear in the first input, C-linear in the second input, hw, vi = hv, wi hv, vi > 0 ∀ v ∈ V −0. and If h, i and h, i′ are nondegenerate Hermitian inner-products on V and a, a′ ∈ R̄+ are not both zero, then ah, i+a′ h, i′ is also a nondegenerate Hermitian inner-product on V . If W is a complex subspace of V and h, i is a nondegenerate Hermitian inner-product on V , let W ⊥ = v ∈ V : hv, wi = 0 ∀ w ∈ W be the orthogonal complement of W in V . In particular, V = W ⊕ W ⊥. Furthermore, the quotient projection map π : V −→ V /W induces an isomorphism from W ⊥ to V /W so that V ≈ W ⊕ (V /W ). If M is a smooth manifold and V −→ M is a smooth complex vector bundle of rank k, a Hermitian metric on V is a nondegenerate Hermitian inner-product in each fiber Vx ≈ Ck of V that varies smoothly with x ∈ M . Formally, the smoothness requirement is one of the following equivalent conditions: 65 (a) the map h, i : V × M V −→ C is smooth; (b) the section h, i of the vector bundle (V ⊗R V )∗ −→ M is smooth; (c) if s1 , s2 are smooth sections of the vector bundle V −→ M , then the function s1 , s2 on M is smooth; (d) if h : V |U −→ U ×Ck is a trivialization of V , then the matrix-valued function, B : U −→ Matk C s.t. h−1 (p, v), h−1 (p, w) = v̄ t B(p)w ∀ p ∈ M, v, w ∈ Ck , is smooth. Similarly to the real case, every complex vector bundle admits a Hermitian metric. If W is a subspace of V and h, i is a Hermitian metric on V , let W ⊥ = v ∈ V : hv, wi = 0 ∀ w ∈ W be the orthogonal complement of W in V . Then W ⊥ −→ M is a complex vector subbundle of V and V = W ⊕ W ⊥. Furthermore, the quotient projection map π : V −→ V /W induces an isomorphism of complex vector bundles over M so that V ≈ W ⊕ (V /W ). If V −→ M is a real vector bundle of rank k with a Riemannian metric h, i or a complex vector bundle of rank k with a Hermitian metric h, i, let SV ≡ v ∈ V : hv, vi = 1 −→ M be the sphere bundle of V . In the real case, the fiber of SV over every point of M is S k−1 . Furthermore, if U is a small open subset of M , then SV |U ≈ U ×S k−1 as bundles over U , i.e. SV is an S k−1 -fiber bundle over M . In the complex case, SV is an S 2k−1 -fiber bundle over M . If V −→ M is a real line bundle (vector bundle of rank one) with a Riemannian metric h, i, then SV −→ M is an S 0 -fiber bundle. In particular, if U is a small open subset of M , SV |U is diffeomorphic to U ×{±1}. Thus, SV −→ M is a 2 : 1-covering map. If M is connected, the covering space SV is connected if and only if V is not orientable; see Section 12 below. 12 Orientations If V is a real vector space of dimension k, the top exterior power of V , i.e. Λtop V ≡ Λk V 66 is a one-dimensional vector space. Thus, Λtop V − 0 has exactly two connected components. An orientation on V is a component C of Λtop V −0. If C is an orientation on V , then a basis {ei } for V is called oriented (with respect to C) if e1 ∧ . . . ∧ ek ∈ C. If {fj } is another basis for V and A is the change-of-basis matrix from {ei } to {fj }, i.e. then f1 , . . . , fk = e1 , . . . , ek A ⇐⇒ fj = i=k X Aij ei , i=1 f1 ∧ . . . ∧ fk = (det A)e1 ∧ . . . ∧ ek . Thus, two different bases for V belong to the same orientation on V if and only if the determinant of the corresponding change-of-basis matrix is positive. Suppose V −→ M is a real vector bundle of rank k. An orientation for V is an orientation for each fiber Vx ≈ Rk , which varies smoothly (or continuously, or is locally constant) with x ∈ M . This means that if h : V |U −→ U ×Rk is a trivialization of V and U is connected, then h is either orientation-preserving or orientationreversing (with respect to the standard orientation of Rk ) on every fiber. If V admits an orientation, V is called orientable. Lemma 12.1. Suppose V −→ M is a smooth real vector bundle. (1) V is orientable if and only if V ∗ is orientable. (2) V is orientable if and only if there exists a collection {Uα , hα } of trivializations that covers M such that det gαβ : Uα ∩Uβ −→ R+ , where {gαβ } is the corresponding transition data. (3) V is orientable if and only if the line bundle Λtop V −→ M is orientable. (4) If V is a line bundle, V is orientable if and only if V is (isomorphic to) the trivial line bundle M ×R. (5) If M is connected and V is a line bundle, V is orientable if and only if the sphere bundle SV (with respect to any Riemann metric on V ) is not connected. Proof: (1) Since Λtop (V ∗ ) ≈ (Λtop V )∗ and a line bundle L is trivial if and only if L∗ is trivial, this claim follows from (3) and (4). (2) If V has an orientation, we can choose a collection {Uα , hα } of trivializations that covers M such that the restriction of hα to each fiber is orientation-preserving (if a trivialization is orientationreversing, simply multiply its first component by −1). Then, the corresponding transition data {gαβ } is orientation-preserving, i.e. det gαβ : Uα ∩Uβ −→ R+ . 67 Conversely, suppose {Uα , hα } is a collection of trivializations that covers M such that det gαβ : Uα ∩Uβ −→ R+ . Then, if x ∈ Uα for some α, define an orientation on Vx by requiring that hα : Vx −→ x×Rk is orientation-preserving. Since det gαβ is R+ -valued, the orientation on Vx is independent of α such that x ∈ Uα . Each of the trivializations hα is then orientation-preserving on each fiber. (3) An orientation for V is the same as an orientation for Λtop V , since Λtop V = Λtop Λtop V . Furthermore, if {(Uα , hα )} is a collection of trivializations for V such that the corresponding transition functions gαβ have positive determinant, then {(Uα , Λtop hα )} is a collection of trivializations for Λtop V such that the corresponding transition functions Λtop gαβ = det(gαβ ) have positive determinant as well. (4) The trivial line bundle M×R is orientable, with an orientation determined by the standard orientation on R. Thus, if V is isomorphic to the trivial line bundle, then V is orientable. Conversely, suppose V is an oriented line bundle. For each x ∈ M , let Cx ⊂ Λtop V = V be the chosen orientation of the fiber. Choose a Riemannian metric on V and define a section s of V by requiring that for all x ∈ M s(x), s(x) = 1 and s(x) ∈ Cx . This section is well-defined and smooth (as can be seen by looking on a trivialization). Since it does not vanish, the line bundle V is trivial by Lemma 8.5. (5) If V is orientable, then V is isomorphic to M ×R, and thus SV = S(M ×R) = M ×S 0 = M ⊔M is not connected. Conversely, if M is connected and SV is not connected, let SV + be one of the components of SV . Since SV −→ M is a covering projection, so is SV + −→ M . Since the latter is one-to-one, it is a diffeomorphism, and its inverse determines a nowhere-zero section of V . Thus, V is isomorphic to the trivial line bundle by Lemma 8.5. If V is a complex vector space of dimension k, V has a canonical orientation as a real vector space of dimension 2k. If {ei } is a basis for V over C, then e1 , ie1 , . . . , ek , iek is a basis for V over R. The orientation determined by such a basis is the canonical orientation for the underlying real vector space V . If {fj } is another basis for V over C, B is the complex change-of-basis matrix from {ei } to {fj }, A is the real change-of-basis matrix from to f1 , if1 , . . . , fk , ifk , e1 , ie1 , . . . , ek , iek 68 then det A = (det B)det B ∈ R+ . Thus, the two bases over R induced by complex bases for V determine the same orientation for V . This implies that every complex vector bundle V −→ M is orientable as a real vector bundle. A smooth manifold M is called orientable if its tangent bundle, T M −→ M , is orientable. Corollary 12.2. Let M be a smooth manifold. The following statements are equivalent: (1) M is orientable; (2) the bundle T ∗ M −→ M is orientable; (3) M admits a volume form; (4) there exists a collection of smooth charts {(Uα , ϕα )}α∈A that covers M such that det J (ϕα ◦ϕ−1 β )x > 0 ∀ x ∈ ϕβ (Uα ∩Uβ ), α, β ∈ A. The equivalence of the first three conditions follows immediately from Lemma 12.1. If {(Uα , ϕα )}α∈A is a collection of charts as in (4), then hα = ϕ̃α : T M |Uα −→ Uα ×Rm , v −→ π(v), v(ϕα ) , is a collection of trivializations of T M as in Lemma 12.1-(2) for V = T M , since m m −1 ϕ̃α ◦ ϕ̃−1 β : Uα ∩Uβ × R −→ Uα ∩Uβ × R , (p, v) −→ p, J (ϕα ◦ϕβ )ϕβ (p) v), m m hα ◦h−1 β : Uα ∩Uβ × R −→ Uα ∩Uβ × R , (p, v) −→ p, gαβ (p)v). In particular, if such a collection of charts exists, then T M is orientable. Conversely, suppose {(Uα , hα )}α∈A is a collection of trivializations of T M as in Lemma 12.1-(2), {(Uα , ϕα )}α∈A is any collection of smooth charts on M , and Uα is connected. In particular, m m ϕ̃α ◦ h−1 (p, v) −→ p, {h−1 α : Uα ×R −→ Uα ×R , α (p, v)}(ϕα ) , is a smooth vector-bundle isomorphism. Thus, there is a smooth map Aα : Uα −→ GLm R s.t. m {h−1 α (p, v)}(ϕα ) = Aα (p)v ∀ v ∈ R . Since Uα is connected, det Aα does not change sign on Uα . By changing the sign of the first component of ϕα if necessary, it can be assumed that det Aα (p) > 0 for all p ∈ Uα and α ∈ A. Thus, −1 det J (ϕα ◦ϕ−1 β )ϕβ (p) = det Aα (p) · det gαβ (p) · det Aβ (p) > 0 ∀ p ∈ Uα ∩Uβ , α, β ∈ A. Thus, the collection {(Uα , ϕα )}α∈A satisfies (4). An orientation for a smooth manifold M is an orientation for the vector bundle T M −→ M ; a manifold with a choice of orientation is called oriented. A diffeomorphism f : M −→ N between oriented manifolds is called orientation-preserving (orientation-reversing) if the differential dp f : Tp M −→ Tf (p) N 69 is an orientation-preserving (orientation-reversing) isomorphism for every p ∈ M ; if M is connected, this is the case if and only if dp f is orientation-preserving (orientation-reversing) for a single point p∈M. If M is a smooth manifold, the sphere bundle π : S Λtop T ∗ M −→ M is a two-to-one covering map. By Lemma 12.1 and Corollary 12.2, if M is connected, the domain of π is connected if and only if M is not orientable. For each p ∈ M , π −1 (p) ≡ {Ωp , −Ωp } ⊂ S Λtop Tp∗ M ⊂ Λtop Tp∗ M is a pair on nonzero top forms on Tp∗ M , which define opposite orientations of Tp M . Thus, S(Λtop T ∗ M ) can be thought as the set of orientations on the fibers of M ; it is called the orientation double cover of M . Smooth maps f, g : M −→ N are called smoothly homotopic if there exists a smooth map H : M ×[0, 1] −→ N s.t. H(p, 0) = f (p), H(p, 1) = g(p) ∀ p ∈ M. Diffeomorphisms f, g : M −→ N are called isotopic if there exists a smooth map H as above such that the map Ht : M −→ N, p −→ (p, t), is a diffeomorphism for every t ∈ [0, 1]. We leave proofs of the following lemmas as an exercise; both can be proved using Corollary 12.2. Lemma 12.3. The orientation double cover of any smooth manifold is orientable. Lemma 12.4. Let f, g : M −→ N be isotopic diffeomorphisms between oriented manifolds. If f is orientation-preserving (orientation-reversing), then so is g. Exercises 1. Let π : V −→ M be a vector bundle. Show that (a) the scalar-multiplication map (7.1) is smooth; (b) the space V ×M V is a smooth submanifold of V ×V and the addition map (7.2) is smooth. 2. Let π : V −→ M be a smooth vector bundle of rank k and {(Uα , hα )}α∈A a collection of trivializations covering M . Show that a section s of π is continuous (smooth) if and only if the map sα ≡ π2 ◦hα ◦s : Uα −→ Rk , where π2 : Uα×Rk −→ Rk is the projection on the second component, is continuous (smooth) for every α ∈ A. 3. Let π : V −→ M be a submersion satisfying (RVB1)-(RVB3) in Definition 7.1. Show that 70 (a) if s1 , . . . , sk : U −→ V |U are smooth sections over an open subset U ⊂ M such that {si (x)}i is a basis for Vx for all x ∈ U , then the map (8.2) is a diffeomorphism; (b) π : V −→ M is a vector bundle of rank k if and only if for every p ∈ M there exist a neighborhood U of p in M and smooth sections s1 , . . . , sk : U −→ V |U such that {si (p)}i is a basis for Vp . 4. Show that the two versions of the last condition on f˜ in (2) in Definition 8.2 are indeed equivalent. 5. Let M be a smooth manifold and X, Y, Z ∈ VF(M ). Show that (a) [X, Y ] is indeed a smooth vector field on M and [f X, gY ] = f g[X, Y ] + f (Xg)Y − g(Y f )X ∀ f, g ∈ C ∞ (M ); (b) [·, ·] is bilinear, anti-symmetric, and X, [Y, Z] + Y, [Z, X] + Z, [X, Y ] = 0. 6. Verify all claims made in Example 7.5, thus establishing that the tangent bundle T M of a smooth manifold is indeed a vector bundle. What is its transition data? 7. Show that the tangent bundle T S 1 of S 1 is isomorphic to the trivial real line bundle over S 1 . 8. Show that the tautological line bundle γn −→ RP n is non-trivial for n ≥ 1. 9. Show that the complex tautological line bundle γn −→ CP n is indeed a complex line bundle as claimed in Example 7.8. What is its transition data? Why is it non-trivial for n ≥ 1? 10. Let q : M̃ −→ M be a smooth covering projection. Show that (a) the map dq : M̃ −→ M is a covering projection and a bundle homomorphism covering q as in (8.4); (b) there is a natural isomorphism VF(M ) ≈ V F (M̃ )dq ≡ X ∈ VF : dp1 q(X(p1 )) = dp2 q(X(p2 )) ∀ p1 , p2 ∈ M s.t. q(p1 ) = q(p2 ) . 11. Let M be a smooth m-manifold. Show that (TM1) the topology on T M constructed in Example 7.5 is the unique one so that π : T M −→ M is a topological vector bundle with the canonical vector-space structure on the fibers and so that for every vector field X on T M and smooth function f : U −→ R, where U is an open subset of R, the function X(f ) : U −→ R is continuous if and only if X is continuous; (TM2) the smooth structure on T M constructed in Example 7.5 is the unique one so that π : T M −→ M is a smooth vector bundle with the canonical vector-space structure on the fibers and so that for every vector field X on T M and smooth function f : U −→ R, where U is an open subset of R, the function X(f ) : U −→ R is smooth if and only if X is smooth. 71 12. Suppose that f : M −→ N is a smooth map and π : V −→ N is a smooth vector bundle of rank k with transition data {gαβ : Uα ∩Uβ −→ GLn R}α,β∈A . Show that (a) the space f ∗ V defined by (10.1) is a smooth submanifold of M × V and the projection π1 : f ∗ V −→ M is a vector bundle of rank k with transition data ∗ f gαβ = gαβ ◦f : f −1 (Uα )∩f −1 (Uβ ) −→ GLn R}α,β∈A ; (b) if M is an embedded submanifold of N and f is the inclusion map, then the projection π2 : f ∗ V −→ V induces an isomorphism f ∗ V −→ V |M of vector bundles over M . 13. Let f : M −→ V be a smooth map and V −→ N a vector bundle. Show that (a) if V −→ N is a trivial vector bundle, then so is f ∗ V −→ M ; (b) f ∗ V −→ M may be trivial even if V −→ N is not. 14. Let f : M −→ N be a smooth map. Show that the bundle homomorphisms in diagrams (10.4) and (10.5) are indeed smooth. 15. Verify Lemma 10.2. 16. Let f : M −→ N be a smooth map and ϕ : V −→ W a smooth vector-bundle homomorphism over N . Show that the pullback vector-bundle homomorphism f ∗ ϕ : f ∗ V −→ f ∗ W is also smooth. 17. Let π : V −→ M be a smooth vector bundle of rank k and V ′ ⊂ V a smooth submanifold so that Vp′ ≡ Vp ∩V ′ is a k′ -dimensional linear subspace of Vp for every p ∈ M . Show that (a) for every p ∈ M = s0 (M ) there exist an open neighborhood U of p in V ′ and smooth charts ϕ : U −→ Rm ×Rk ′ and ψ : U ∩M −→ Rm s.t. ψ ◦ π = π1 ◦ ϕ , ′ where π1 : Rm ×Rk −→ Rm is the projection on the first component; (b) V ′ ⊂ V is a vector subbundle of rank k′ . 18. Let ϕ : V −→ W be a smooth surjective vector-bundle homomorphism over a smooth manifold M . Show that ker ϕ ≡ v ∈ V : ϕ(v) = 0 −→ M is a subbundle of V . 19. Let D ⊂ T M a rank 1 distribution on a smooth manifold M . Show that Γ(M ; D) ⊂ VF(M ) is a Lie subalgebra. Hint: use Exercise 5. 20. Let {ια : Mα −→ N }α∈A be a foliation of N n by immersions from m-manifolds. Show that [ [ Im dp ια ⊂ T N D≡ α∈A p∈Mα is a subbundle of rank m. 21. Verify all claims made in Examples 10.6 and 10.7. 72 22. Verify all claims made in Example 10.8. 23. Let π : V −→ M be a smooth vector bundle. Show that (a) the fibers of π form a proper foliation of V ; (b) the corresponding subbundle D ⊂ T V is isomorphic to π ∗ V as vector bundles over V ; (c) there is a short exact sequence of vector bundles dπ 0 −→ π ∗ V −→ T V −→ π ∗ T M −→ 0 24. Let V −→ M be a vector bundle of rank k and V ′ ⊂ V a smooth subbundle of rank k′ . Show that (a) there exists a collection {(Uα , hα )}α∈A of trivializations for V covering M so that (10.9) holds and thus the corresponding transition data has the form ∗ ∗ gαβ = : Uα ∩ Uβ −→ GLk R, 0 ∗ where the top left block is k′ ×k′ ; (b) the vector-bundle structure on V /V ′ described in Section 10 is the unique one so that the natural projection map V −→ V /V ′ is a smooth vector-bundle homomorphism; (c) if ϕ : V −→ W is a vector-bundle homomorphism over M such that ϕ(v) = 0 for all v ∈ V ′ , then the induced vector-bundle homomorphism ϕ̄ : V /V ′ −→ W is smooth. 25. Verify Lemmas 10.9 and 10.10. 26. Obtain Corollary 10.12 from Theorem 10.11. 27. Let f = (f1 , . . . , fk ) : Rm −→ Rk be a smooth map, q ∈ Rk a regular value of f , and X = f −1 (q). Denote by ∇fi the gradient of fi . Show that T X = (p, v) ∈ X ×Rm : ∇fi |p ·v = 0 ∀ i = 1, 2, . . . , k under the canonical identifications T X ⊂ T Rm |X and T Rm = Rm ×Rm . Use this description of T X to give a trivialization of NRm X. 28. Let V, V ′ −→ M be smooth vector bundles. Show that the two constructions of V ⊕ V ′ in Section 10 produce the same vector bundle and that this is the unique vector-bundle structure on the total space of G V ⊕V ′ = Vp ⊕Vp′ p∈M so that L (VB 1) the projection maps V ⊕V ′ −→ V, V ′ are smooth bundle homomorphisms over M ; L (VB 2) the inclusion maps V, V ′ −→ V ⊕V ′ are smooth bundle homomorphisms over M . 29. Let πV : V −→ M and πW : W −→ N be smooth vector bundles and πM , πN : M ×N −→ M, N the component projection maps. Show that the total of the vector bundle ∗ ∗ π : πM V ⊕ πN W −→ M ×N is V ×W (with the product smooth structure) and π = πV ×πW . 73 30. Verify Lemmas 10.13 and 10.14. 31. Let M and N be smooth manifolds and πM , πN : M ×N −→ M, N the projection maps. Show that dπM and dπN viewed as maps from T (M ×N ) to (a) T M and T N , respectively, induce a diffeomorphism T (M×N ) −→ T M×T N that commutes with the projections from the tangent bundles to the manifolds and is linear on the fibers of these projections; ∗ T M and π ∗ T N , respectively, induce a vector-bundle isomorphism (b) πM N ∗ ∗ T (M ×N ) −→ πM T M ⊕πN T N. Why are the above two statements the same? 32. Verify Lemmas 10.15 and 10.16. 33. Show that the vector-bundle structure on the total space of V ∗ constructed in Section 10 is the unique one so that the map (10.15) is smooth. 34. Verify Lemmas 10.17-10.19. 35. Let V −→ M be a smooth vector bundle of rank k and W ⊂ V a smooth subbundle of V of rank k′ . Show that Ann(W ) ≡ α ∈ Vp∗ : α(w) = 0 ∀ w ∈ W, p ∈ M is a smooth subbundle of V ∗ of rank k−k′ . 36. Verify Lemmas 10.20-10.23. 37. Let π : V −→ M be a vector bundle. Show that there is an isomorphism Λk (V ∗ ) −→ Λk V )∗ of vector bundles over M . 38. Let Ω be a volume form an m-manifold M . Show that for every p ∈ M there exists a chart (x1 , . . . , xm ) : U −→ Rm around p such that Ω|U = dx1 ∧ . . . ∧ dxm . 39. Show that every complex vector bundle V −→ M admits a Hermitian metric. 40. Let π : L −→ M be a real line bundle over a smooth manifold. Show that L⊗2 ≈ τ1R as real line bundles over M . 41. Let V, W −→ M be vector bundles. Show that (a) if V is orientable, then W is orientable if and only if V ⊕W is; (b) if V and W are non-orientable, then V ⊕W may be orientable or non-orientable. 42. Let M be a connected manifold. Show that every real line bundle L −→ M is orientable if and only if π1 (M ) contains no subgroup of index 2. 74 43. Let M and N be nonempty smooth manifolds. Show that M ×N is orientable if and only if M and N are. 44. (a) Let ϕ : M −→ RN be an immersion. Show that M is orientable if and only if the normal bundle to the immersion ϕ is orientable. (b) Show that the unit sphere S n with its natural smooth structure is orientable. 45. Verify Lemmas 12.3 and 12.4. 46. (a) Show that the antipodal map on S n ⊂ Rn+1 (i.e. x −→ −x) is orientation-preserving if n is odd and orientation-reversing if n is even. (b) Show that RP n is orientable if and only if n is odd. (c) Describe the orientable double cover of RP n ×RP n with n even. 47. Let γn −→ CP n be the tautological line bundle as in Example 7.8. If P : Cn+1 −→ C is a homogeneous polynomial of degree d ≥ 0, let sP : CP n −→ γn∗ , sP (ℓ) (ℓ, v ⊗d ) = P (v) ∀ (ℓ, v) ∈ γn ⊂ CP n ×Cn+1 . Show that (a) sP is a well-defined holomorphic section of γn∗⊗d ; (b) if s is a holomorphic section of γn∗⊗d with d ≥ 0, then s = sP for some homogeneous polynomial P : Cn+1 −→ C of degree d; (c) the line bundle γn⊗ −→ CP n admits no nonzero holomorphic section for any d ∈ Z+ . 48. Let γn −→ CP n be the tautological line bundle as in Example 7.8. Show that there is a short exact sequence 0 −→ CP n ×C −→ (n+1)γn∗ −→ T CP n −→ 0 of complex (even holomorphic) vector bundles over CP n . 49. Suppose k < n and let γk −→ CP k be the tautological line bundle as in Example 7.8. Show that the map ι : CP k −→ CP n , [X0 , . . . , Xk ] −→ [X0 , . . . , Xk , 0, . . . , 0], | {z } n−k is a complex embedding (i.e. a smooth embedding that induces holomorphic maps between the charts that determine the complex structures on CP k and CP n ) and that the normal bundle to this immersion, Nι , is isomorphic to (n−k)γk∗ ≡ γk∗ ⊕ . . . ⊕ γk∗ {z } | n−k as a complex (even holomorphic) vector bundle over CP k . Hint: there are a number of ways of doing this, including: (i) use Exercise 48; (ii) construct an isomorphism between the two vector bundles; 75 (iii) determine transition data for Nι and (n−k)γk∗ ; (iv) show that there exists a holomorphic diffeomorphism between (n − k)γk∗ and a neighborhood of ι(CP k ) in CP n , fixing ι(CP k ), and that this implies that Nι = (n−k)γk∗ . 50. Let γn −→ CP n and ΛnC T CP n −→ CP n be the tautological line bundle as in Example 7.8 and the top exterior power of the vector bundle T CP n taken over C, respectively. Show that there is an isomorphism ΛnC T CP n ≈ γn∗ ⊗(n+1) ≡ γn∗ ⊗ . . . ⊗ γn∗ | {z } n+1 of complex (even holomorphic) line bundles over 76 CP n . Hint: see suggestions for Exercise 49. Chapter 3 Frobenius Theorems 13 Integral Curves Recall from Section 8 that a vector field X on a smooth manifold M is a section of the tangent bundle T M −→ M . Thus, X : M −→ T M is a map such that X(p) ∈ Tp M for all p ∈ M . If ϕ = (x1 , . . . , xm ) : U −→ M is a smooth chart on M , then i=m X ∂ ∀ p ∈ U, ci (p) X(p) = ∂xi p i=1 for some functions c1 , . . . , cm : U −→ R. The vector field X is smooth (as a map between the smooth manifolds M and T M ) if and only if the functions c1 , . . . , cm corresponding to every smooth chart on M are smooth. This is the case if and only if X(f ) : M −→ R is smooth function for every f ∈ C ∞ (M ). As defined in Section 4, a smooth curve on M is a smooth map γ : (a, b) −→ M . For t ∈ (a, b), the tangent vector to a smooth curve γ at t is the vector d γ ′ (t) = γ(t) ≡ dt γ ∂e1 |t ∈ Tγ(t) M, dt 1 where e1 = 1 ∈ R is the oriented unit vector. Definition 13.1. Let X be a smooth vector field on a smooth manifold M . An integral curve for X is a smooth curve γ : (a, b) −→ M γ ′ (t) = X(γ(t)) ∀ t ∈ (a, b). s.t. For example, a smooth vector field X on R2 has the form ∂ ∂ X(x, y) = f (x, y) + g(x, y) ∂x (x,y) ∂y (x,y) (13.1) for some f, g ∈ C ∞ (R2 ). A smooth map γ = (γ1 , γ2 ) : (a, b) −→ R2 is an integral curve for such a vector field if ∂γ2 (t) ∂ ∂γ1 (t) ∂ ( + γ ′ (t) = ∂t ∂x γ(t) ∂t ∂y γ(t) γ1′ (t) = f γ1 (t), γ2 (t) ⇐⇒ ∂ ∂ γ2′ (t) = g γ1 (t), γ2 (t) + g(γ(t)) = f (γ(t)) ∂x ∂y (γ(t)) (γ(t)) 77 This is a system of two ordinary autonomous first-order differential equations for γ = (γ1 , γ2 ) as a function of t. Lemma 13.2. Let X be a smooth vector field on a smooth manifold M . For every p ∈ M , there exists an integral curve γ : (−ǫ, ǫ) −→ M for X such that γ(0) = p. If γ, γ̃ : (−ǫ, ǫ) −→ M are two such integral curves, then γ = γ̃. Let ϕ = (x1 , . . . , xm ) : U −→ Rm be a smooth chart on M around p and c1 , . . . , cm : U −→ R smooth functions such that i=m X ′ ∂ ′ ci (p ) X(p ) = ∀ p′ ∈ U. ∂xi p′ i=1 For any smooth map γ : (a, b) −→ U ⊂ M , let (γ1 , . . . , γm ) = ϕ◦γ : (a, b) −→ Rm . By the chain rule (4.5) and the definition of the coordinate vector fields (4.12), the condition (13.1) on γ is then equivalent to X ∂ i=m ci γ(t) {ϕ◦γ} (t) = dt ϕ γ (t) = ∂xi ϕ(γ(t)) ′ ′ i=1 ⇐⇒ γi′ (t) = ci ◦ϕ−1 γ1 (t), . . . , γm (t) ∀ i. Since the functions ci ◦ϕ−1 are smooth on Rm , the initial-value problem ( γi′ (t) = ci ◦ϕ−1 γ1 (t), . . . , γm (t) i = 1, 2, . . . , m γ1 (0), . . . , γm (0) = ϕ(p) (13.2) has a solution (γ1 , . . . , γm ) : (−ǫ, ǫ) −→ Rm for some ǫ > 0 by the Existence Theorem for FirstOrder Differential Equations [1, A.2]. By the Uniqueness Theorem for First-Order Differential Equations [1, A.1], any two solutions of this initial-value problem must agree on the intersection of the domains of their definition. Corollary 13.3. Let X be a smooth vector field on a smooth manifold M and p ∈ M . If a, ã ∈ R− , b, b̃ ∈ R+ , and γ : (a, b) −→ M and γ̃ : (ã, b̃) −→ M are integral curves for X, then γ|(a,b)∩(ã,b̃) = γ̃|(a,b)∩(ã,b̃) . The subset A ≡ t ∈ (a, b) ∩ (ã, b̃) : γ(t) = γ̃(t) ⊂ (a, b) ∩ (ã, b̃) is nonempty (as it contains 0) and closed (as γ and γ̃ are continuous). Since (a, b) ∩ (ã, b̃) is connected, it is sufficient to show that S is open. If t0 ∈ S (t0 −ǫ, t0 +ǫ) ⊂ (a, b) ∩ (ã, b̃), and define smooth curves α, β : (−ǫ, ǫ) −→ M by α(t) = γ(t+t0 ), β(t) = γ̃(t+t0 ). Since γ and γ̃ are integral curves for X, α′ (t) = d γ(t+t0 ) = X γ(t+t0 ) = X α(t) , dt β ′ (t) = 78 d γ̃(t+t0 ) = X γ̃(t+t0 ) = X β(t) . dt Thus, α and β are integral curves for X. Since α(0) = γ(t0 ) = γ̃(t0 ) = β(0), α = β by Lemma 13.2 and thus (t0 −ǫ, t0 +ǫ) ⊂ (a, b) ∩ (ã, b̃). Corollary 13.4. Let X be a smooth vector field on a smooth manifold M . For every p ∈ M , there exists a unique maximal integral curve γp : (ap , bp ) −→ M for X such that γp (0) = p, where ap ∈ [−∞, 0) and bp ∈ (0, ∞]. If t ∈ (ap , bp ), then γγp (t) (−t) = p. (13.3) aγp (t) , bγp (t) = (ap −t, bp −t), (1) Let {γα : (aα , bα ) −→ M }α∈A be the collection of all integral curves for X such that γα (0) = p. Define [ (ap , bp ) = (aα , bα ), γp : (ap , bp ) −→ M, γp (t) = γα (t) ∀ t ∈ (aα , bα ), α ∈ A. α∈A By Corollary 13.3, γα (t) is independent of the choice of α ∈ A such that t ∈ (aα , bα ). Thus, γp is well-defined. It is smooth, since its restriction to each open subset (aα , bα ) is smooth and these subsets cover (ap , bp ). It is an integral curve for X, since this is the case on the open subsets (aα , bα ). It is immediate that γp (0) = p. By construction, γp is a maximal integral curve for X. (2) If t ∈ (ap , bp ), define γ : (ap −t, bp −t) −→ M by γ(τ ) = γp (τ +t). This is a smooth map such that γ(0) = γp (t), γ(−t) = γp (0) = p, γ ′ (τ ) = d γp (τ +t) = X γp (τ +t) = X γ(t) ; dτ the second-to-last equality above holds because γp is an integral curve for X. Thus, γ is an integral curve for X such that γ(0) = γp (t). In particular, by the first statement of Corollary 13.4, aγp (t) , bγp (t) ⊃ (ap −t, bp −t), γγp (t) (ap −t,bp −t) = γ =⇒ − t ∈ aγp (t) , bγp (t) , γγp (t) (−t) = p =⇒ (ap , bp ) = aγγp (t) (−t) , bγγp (t) (−t) ⊃ aγp (t) +t, bγp (t) +t . This confirms (13.3). If X is a smooth vector field on M , for each t ∈ R let Domt (X) = p ∈ M : t ∈ (ap , bp ) , Xt : Domt (X) −→ M, Xt (p) = γp (t). The map Xt is called the time t flow of vector field X. Example 13.5. Let X be the smooth vector field on M = R given by 2 ∂ . X(x) = −x ∂x x 79 If p ∈ R, the integral curve γp for X is described by γp′ (t) = −γp (t)2 , γp (0) = p ⇐⇒ γp (t) = p . 1 + pt Thus, (−∞, −1/t), Domt (X) = (−∞, ∞), (−1/t, ∞), if t < 0; if t = 0; if t > 0; Xt : Domt (X) −→ R, Xt (p) = p . 1 + tp Example 13.6. Let q : R −→ S 1 , θ −→ e2πiθ , be the usual covering map and X the vector field on S 1 defined by X e2πiθ = dθ q(e1 ), where e1 = 1 is the usual oriented unit vector in R = Tθ R. If q(θ1 ) = q(θ2 ), there exists n ∈ Z such that θ2 = θ1 +n. Define hn : R −→ R by θ −→ θ + n. Since dθ1 h(e1 ) = e1 and q = q◦hn , by the chain rule (4.5) dθ2 q(e1 ) = dθ2 q dθ1 hn (e1 ) = dθ1 {q◦hn }(e1 ) = dθ1 q(e1 ). Thus, the vector field X is well-defined (the value of X at e2πiθ depends only on e2πiθ , and not on θ). This vector field is smooth, since e1 defines a smooth vector field on R, while q : R −→ S 1 and dq : T R −→ T S 1 are covering projections (and in particular local diffeomorphisms). If p ∈ S 1 , p̃ ∈ q −1 (p) ⊂ R, and γ : (a, b) −→ S 1 is a smooth curve such that γ(0) = p, let γ̃ : (a, b) −→ R be the continuous lift of γ over q such that γ̃(0) = p̃; since q is a local diffeomorphism, this map is smooth. The integral curve γp is then described by γ̃p′ (t) = 1, ⇐⇒ γ̃p (0) = p̃ =⇒ γ̃p (t) = p̃ + t ∈ R γp (t) = q γ̃p (t) = e2πi(p̃+t) = e2πip̃ · e2πit = e2πit · p ∈ S 1 . Thus, γp (t) is defined for all t ∈ R, and the time t flow of X is given by Xt : Domt (X) = S 1 −→ S 1 , p −→ e2πit p. This is the rotation by the angle 2πt. Proposition 13.7. If X is a smooth vector field on a smooth manifold M , then (1) Dom0 (X) = M , X0 = idM , and M= [ Domt (X) = t>0 [ Domt (X) ; t<0 Doms (X) −→ M ; (2) for all s, t ∈ R, Xs+t = Xs ◦Xt : Dom(Xs ◦Xt ) = X−1 t (3) for all p ∈ M , there exist an open neighborhood U of p in M and ǫ ∈ R+ such that the map (t, p′ ) −→ Xt (p′ ) ≡ γp′ (t), X : (−ǫ, ǫ)×U −→ M, is defined and smooth; 80 (13.4) (4) for all t ∈ R, Dom(Xt ) ⊂ M is an open subset; (5) for all t ∈ R, Xt : Domt (X) −→ Dom−t (X) is a diffeomorphism with inverse X−t . Proof: (1) By Lemma 13.2, for each p ∈ M there exists an integral curve γ : (−ǫ, ǫ) −→ M for X such that γ(0) = p. Thus, p ∈ Dom±ǫ/2 (X) ⊂ Dom0 (X); this implies the first and last claims in (1). The middle claim follows from the requirement that X0 (p) ≡ γp (0) = p for all p ∈ M . −1 (2) Since Dom(Xs ) = Doms (X), Dom(Xs ◦Xt ) = X−1 t (Doms (X)). If p ∈ Xt (Doms (X)), s ∈ aXt (p) , bXt (p) = aγp (t) , bγp (t) . 1 Thus, s+t ∈ (ap , bp ) by (13.3) and X−1 t (Doms (X)) ⊂ Dom(Xs+t ). Define γ : aXt (p) , bXt (p) −→ M by γ(τ ) = γp (τ +t); by (13.3), γp (τ +t) is defined for all τ ∈ (aXt (p) , bXt (p) ). The map γ is smooth and satisfies γ(0) = γp (t) = Xt (p), γ ′ (τ ) = d γp (τ +t) = X γp (τ +t) = X γ(τ ) ; dτ the second-to-last equality holds because γp is an integral curve for X. Thus, by Corollary 13.4, γ = γXt (p) and so Xs+t (p) ≡ γp (s+t) = γ(s) = γX(t) (s) ≡ Xs Xt (p) for all s ∈ (aXt (p) , bXt (p) ). (3) As in the proof of Lemma 13.2, the requirement for a smooth map γ : (a, b) −→ M to be an integral curve for X passing through p corresponds to an initial-value problem (13.2) in a smooth chart around p. Thus, the claim follows from the smooth dependence of solutions of (13.2) on the parameters [1, A.4]. (4) Since Dom0 (X) = M and Dom−t (X) = Domt (−X), it is sufficient to prove this statement for t ∈ R+ . Let p ∈ Domt (X) and W ⊂ M be an open neighborhood of Xt (p) = γp (t) in M . Since the interval [0, t] is compact, by (3) and Lebesgue Number Lemma (Lemma B.1.2), there exist ǫ > 0 and a neighborhood U of γp ([0, t]) such that the map (13.4) is defined and smooth. Let n ∈ Z+ be such that t/n < ǫ. We inductively define subsets Wi ⊂ M by Wn = W, −1 Wi = X−1 (Wi+1 ) t/n (Wi+1 ) ∩ U = Xt/n |U ∀ i = 0, 1, . . . , n−1. By induction, Wi ⊂ U is an open neighborhood of γp (it/n), Wi ⊂ X−1 t/n (Dom(X(n−1−i)t/n )), and thus X(n−i)t/n = Xt/n ◦ X(n−1−i)t/n : Wi −→ U ⊂ M by (2). It follows that W0 ⊂ M is an open neighborhood of p in M such that W0 ⊂ Domt (X). (5) By (13.3) and (2), Im Xt = Dom−t (X) and X−t is the inverse of Xt . If p ∈ Domt (X) and W0 is a neighborhood of p in M as in the proof of (4), Xt |W0 is a smooth map. Thus, Xt is smooth on the open subset Domt (X) ⊂ M . 1 The domain of Xs+t might be larger than Dom(Xs ◦Xt ). For example, if s = −t, the former is all of M . 81 Rm−1 R1 Figure 3.1: Flows of a nonvanishing vector field and integral immersions of a rank 1 distribution are horizontal slices in a coordinate chart. Lemma 13.8. Let X be a smooth vector field on a smooth manifold M and p ∈ M . If X(p) 6= 0, there exists a smooth chart ϕ ≡ (x1 , . . . , xm ) : U −→ (−δ, δ)×Rm−1 around p on M such that ∂ X(p ) = ∂x1 p′ ′ ∀ p′ ∈ U. (13.5) By Proposition 13.7-(3), there exist an open neighborhood U of p in M and ǫ ∈ R+ such that the map (13.4) is smooth. Let U ′ be a neighborhood of p in U and φ ≡ (y1 , . . . , ym ) : (U ′ , p) −→ (Rm , 0) a smooth chart such that (13.5) holds with x1 replaced by y1 for p′ = p; such a chart can be obtained by composing another chart with a rigid transformation of Rm . Define ψ : (−ǫ, ǫ)×Rm−1 −→ M by ψ(x1 , x2 , . . . , xm ) = Xx1 φ−1 (0, x2 , . . . , xm ) . This smooth map sends (x1 , x2 , . . . , xm ) to the time x1 -flow from the point φ−1 (0, x2 , . . . , xm ) on the coordinate hyperplane φ−1 (0×Rm−1 ); see Figure 3.1. Note that d ∂ d ; = Xt (p) = X(p) = d0 ψ ∂e1 φ(p) = ψ(t, 0, . . . , 0) dt dt ∂y1 p t=0 t=0 d −1 ∂ d ∀ i ≥ 2; = φ (0, . . . , 0, t, 0, . . . , 0) = d0 ψ ∂ei φ(p) = ψ(0, . . . , 0, t, 0, . . . , 0) dt dt ∂yi p t=0 t=0 on the second line, t is inserted into the i-th slot. Thus, the differential of ψ at 0, d0 ψ : T0 Rm −→ Tp M is an isomorphism. By the Inverse Function Theorem for Manifolds (Corollary 4.9), there are neighborhoods U of p in M and V of 0 in Rm such that ψ : V −→ U is a diffeomorphism. The inverse of this diffeomorphism is a smooth chart around p on M satisfying (13.5). Corollary 13.9. If D ⊂ T M is a rank 1 distribution on a smooth manifold M , there exists a foliation {ια : R −→ M }α∈A by injective immersions integral to the distribution D on M . 82 Let h : D|W −→ W ×R be a trivialization of D over an open subset W ⊂ M and X(p) = h−1 (p, 1) ∈ Dp ⊂ Tp M ∀ p ∈ W. Since h is a smooth, X is a smooth nowhere 0 vector field on the open subset W ⊂ M . Since the rank of D is 1, Dp = RX(p) for all p ∈ W . Let ϕ ≡ (x1 , . . . , xm ) : U −→ (−δ, δ)×Rm−1 be a coordinate chart on W ⊂ M satisfying (13.5) and φ : R −→ (−δ, δ) any diffeomorphism. For y ∈ Rm−1 , define ιy : R −→ M by ιy (t) = ϕ−1 φ(t), 0, . . . , 0 . This is an injective immersion such that Im ιy is contained in (in fact is) the horizontal slice ϕ−1 (R×y) and ∂ = R X ιy (t) = Dιy (t) ∀ t ∈ R. Im dt ιy = R ∂x1 ιy (t) Thus, {ιy }y∈Rm−1 is a foliation of the open subset U ⊂ M by immersions integral to D. The flows of a vector field X provide a way of differentiating other vector fields and differential forms in the direction of X. Definition 13.10. Let X be a smooth vector field on a smooth manifold M and p ∈ M . (1) The Lie derivative of a smooth vector field Y ∈ VF(M ) on M with respect to X at p is the vector dXt (p) X−t Y (Xt (p)) − Y (p) d ∈ Tp M . (LX Y )p = dXt (p) X−t Y (Xt (p)) ≡ lim t−→0 dt t t=0 (2) The Lie derivative of a smooth k-form α ∈ E k (M ) on M with respect to X at p is the alternating k-tensor X∗t α Xt (p)) − α(p) d ∗ (LX Y )p = Xt α Xt (p)) ∈ Λk Tp∗ M . ≡ lim t−→0 dt t t=0 Thus, the Lie derivative LX measures the rate of change of a smooth vector field Y at p by bringing Y (Xt (p)) ∈ TXt (p) M back to Tp M by the differential of the inverse flow X−t . Similarly, LX measures the rate of change of a smooth k-form α at p by pulling α(Xt (p)) ∈ Λk (TX∗t (p) M ) back to Λk (Tp∗ M ) by X∗t = Λk dp Xt ∗ : Λk (TX∗t (p) M ) −→ Λk (Tp∗ M ). As indicated by the following proposition, (LX Y )p and (LX α)p typically depend on the germ of X at p, and not just on X(p). Proposition 13.11. Let X be a smooth vector field on a smooth manifold M and p ∈ M . (1) If f ∈ C ∞ (M ), (LX f )p = Xp (f ). (2) If Y ∈ VF(M ), (LX Y )p = [X, Y ]. 83 (3) If α ∈ E k (M ) and Y1 , Y2 , . . . , Yk ∈ VF(M ), LX α(Y1 , . . . , Yk ) p = {LX α}p Y1 (p), . . . , Yk (p) + i=k X i=1 αp Y1 (p), . . . , Yi−1 (p), (LX Yi )p , Yi+1 (p), . . . , Yk (p) . Corollary 13.12. If X, Y ∈ VF(M ) are smooth vector fields on a smooth manifold M , L[X,Y ] = LX , LY ≡ LX ◦ LY − LY ◦ LX : VF(M ) −→ VF(M ), E k (M ) −→ E k (M ). Exercises 1. Let V be the vector field on R3 given by V (x, y, z) = y ∂ ∂ ∂ −x + . ∂x ∂y ∂z Explicitly describe and sketch the flow of V . 2. Let X be a smooth vector field on a manifold M . Show that (a) if γ : (a, b) −→ M is an integral curve for X such that γ ′ (t) = 0 for some t ∈ (a, b), then γ is a constant map. (b) if X is compactly supported, i.e. supp X ≡ p ∈ M : Xp 6= 0 is a compact subset of M , then Domt (X) = M for all t ∈ R. 3. (a) Let M be a smooth compact manifold and X ∈ VF(M ) a nowhere-zero vector field on M , i.e. X(p) 6= 0 for all p ∈ M . Show that the flow Xt : M −→ M of X has no fixed points for some t ∈ R. (b) Show that S n admits a smooth nowhere-zero vector field if and only if n is odd. Hint: Exercises 46 and 12 in Chapter 2 might be helpful for n even. (c) Show that the tangent bundle of S n is not trivial if n ≥ 1 is even. (In fact, T S n is trivial if and only if n = 1, 3, 7 [2].) 4. Let γ : (a, b) −→ R2 be an integral curve for a smooth vector field X on R2 . Show that γ is an embedding. 5. Let X be a smooth vector field on a smooth manifold M . Show that (a) for t ∈ R, Dom−t (X) = Domt (−X); (b) if s, t ∈ R have the same sign, then Dom(Xs+t ) = Dom(Xs ◦Xt ); (c) if Domt (X) = M for some t ∈ R+ , then Domt (X) = M for all t ∈ R+ . 84 6. Suppose X and Y are smooth vector fields on a manifold M . Show that for every p ∈ M and f ∈ C ∞ (M ), f Y−s (X−t Ys (Xt (p))) − f (p) lim = [X, Y ]p f ∈ R. s,t−→0 st Do not forget to explain why the limit exists. 7. Let X be the vector field on Rn given by X = i=n X i=1 (a) Determine the time t-flow Xt : Rn −→ Rn xi ∂ . ∂xi of X (give a formula). (b) Use (a) to show directly from the definition of the Lie derivative LX that the homomorphism defined by Z 1 k−1 k n k n s f (sx)ds dxi1 ∧ . . . ∧ dxik Rk : E (R ) −→ E (R ), f dxi1 ∧ . . . ∧ dxik −→ 0 is a left inverse for LX if k ≥ 1. (c) Is Rk also a right inverse for LX for k ≥ 1? What happens for k = 0? 8. Verify Corollary 13.12. 9. Let U and V be the vector fields on R3 given by U (x, y, z) = ∂ ∂x and V (x, y, z) = F (x, y, z) ∂ ∂ + G(x, y, z) , ∂y ∂z where F and G are smooth functions on R3 . Show that there exists a proper foliation of R3 by 2-dimensional embedded submanifolds such that the vector fields U and V everywhere span the tangent spaces of these submanifolds if and only if F (x, y, z) = f (y, z) eh(x,y,z) and G(x, y, z) = g(y, z) eh(x,y,z) for some f, g ∈ C ∞ (R2 ) and h ∈ C ∞ (R3 ) such that (f, g) does not vanish on R2 . 10. Let α be a k-form on a smooth manifold M and X0 , . . . , Xk ∈ VF(M ). Show that i=k X ci , . . . , Xk ) (−1)i Xi α(X0 , . . . , X dα(X0 , . . . , Xk ) = i=0 + X ci , . . . , X cj , . . . , Xk . (−1)i+j α [Xi , Xj ], X0 , . . . , X i<j Hint: first show that the values of both sides at any point p ∈ M depend only on the values of vector fields Xi |p at p and on the restriction α|U of α to any neighborhood U of p; then compute in a smooth chart. 11. Let α be a nowhere-zero closed (m−1)-form on an m-manifold M . Show that for every p ∈ M there exists a chart (x1 , . . . , xm ) : U −→ Rm around p such that α|U = dx2 ∧ dx3 ∧ . . . ∧ dxm . Hint: Exercises 38 in Chapter 2 and 3 in Appendix A might be helpful. 85 12. Let ω be a smooth closed everywhere nondegenerate2 two-form on a smooth manifold M . (a) Show that the dimension of M is even and the map T M −→ T ∗ M, X −→ iX ω, is a vector-bundle isomorphism (iX is the contraction w.r.t. X, i.e. the dual of X∧). (b) If H : M −→ R is a smooth map, let XH ∈ VF(M ) be the preimage of dH under this isomorphism. Assume that the flow ϕ : R×M −→ M, (t, p) −→ ϕt (p) of XH is defined for all (t, p). Show that for every t ∈ R, the time-t flow ϕt : M −→ M is a symplectomorphism, i.e. ϕ∗t ω = ω. 13. Suppose M is a 3-manifold, α is a nowhere-zero one-form on M , and p ∈ M . Show that (a) if there exists an embedded 2-dimensional submanifold P ⊂ M such that p ∈ P and α|T P = 0, then (α ∧ dα)|p = 0. (b) if there exists a neighborhood U of p in M such that (α ∧ dα)|U = 0, then there exists an embedded 2-dimensional submanifold P ⊂ M such that p ∈ P and α|T P = 0. 14. Let α = dx1 + f dx2 be a smooth 1-form on R3 (so f ∈ C ∞ (R3 )). Show that for every p ∈ R3 there exists a diffeomorphism ϕ = (y1 , y2 , y3 ) : U −→ V from a neighborhood U of p to an open subset V of R3 such that α|U = gdy1 for some g ∈ C ∞ (U ) if and only if f does not depend on x3 . 15. Let X be a non-vanishing vector field on R3 , written in coordinates as X(x, y, z) = f ∂ ∂ ∂ +g +h ∂x ∂y ∂z for some f, g, h ∈ C ∞ (R3 ). (a) Find a one-form α on R3 so that at each point of R3 the kernel of α is orthogonal to X, with respect to the standard inner-product on R3 . (b) Find a necessary and sufficient condition on X so that for every point p ∈ R3 there exists a surface S ⊂ R3 passing through p which is everywhere orthogonal to X (i.e. S is a smooth two-dimensional submanifold of R3 and Tq S ⊂ Tq R3 is orthogonal to X(q) for all q ∈ S). 2 This means that ωp ∈ Λ2 Tp∗ M is nondegenerate for every p ∈ M , i.e. for every v ∈ Tp M −0 there exists v ′ ∈ Tp M such that ωp (v, v ′ ) 6= 0. 86 Appendix A Linear Algebra A.1 Exercises 1. Let V be a finite-dimensional vector space and v ∈ V −0. Show that (a) if w ∈ Λk V , v∧w = 0 ∈ Λk+1 V if and only if w = v∧u for some u ∈ Λk−1 V ; (b) the sequence of vector spaces v∧· v∧· v∧· 0 −→ Λ0 V −→ Λ1 V −→ Λ2 V −→ . . . is exact. 2. Let V be a vector space of dimension n and ω ∈ Λ2 V an element such that ω n 6= 0 ∈ Λ2n V . Show that the homomorphism ω k ∧ · : Λn−k V −→ Λn+k V, w −→ ω k ∧ w, is an isomorphism for all k ∈ Z+ . 3. Let V be a vector space of dimension n and Ω ∈ Λn V ∗ a nonzero element. Show that the homomorphism V −→ Λn−1 V ∗ , v −→ iv Ω, where iv is the contraction map, is an isomorphism. 4. Show that every short exact sequence of vector spaces, f f 0 −→ A −→ B −→ C −→ 0 induces a canonical isomorphism Λtop A ⊗ Λtop C −→ Λtop B (the isomorphism is determined by f and g). 5. Let {e1 , . . . , ek } and {f1 , . . . , fk } be C-bases for a vector space V over C. Let A be the complex change-of-basis matrix from {ei } to {fj } and B the real change-of-basis matrix from {e1 , ie1 , . . . , ek , iek } to {f1 , if1 , . . . , fk , ifk }. Show that det B = (det A)det A. 92 Appendix B Topology B.1 Lemma B.1.1. Let M be a set and {ϕα : Uα −→ Mα α∈A a collection of bijections from subsets Uα of M to topological spaces Mα such that ϕα ◦ϕ−1 β : ϕβ Uα ∩Uβ −→ ϕα Uα ∩Uβ is a homeomorphism between open subsets of Mβ and Mα , respectively, for all α, β ∈ A. If the collection {Uα }α∈A covers M , then M admits a unique topology TM such that each map ϕα is a homeomorphism. If in addition (1) the collection {Uα }α∈A separates points in M , then the topology TM is Hausdorff; (2) there exists a countable subset A0 ⊂ A such that the collection {Uα }α∈A0 covers M and Mα is second-countable for all α ∈ A0 , then the topology TM is second-countable. A basis for the topology TM consists of the subsets U ⊂ M such that U ⊂ Uα and ϕα (U ) ⊂ Mα is open for some α ∈ A. Lemma B.1.2 (Lebesgue Number Lemma,[7, Lemma 27.5]). Let (M, d) be a compact metric space. For every open cover {Uα }α∈A of M , there exists δ ∈ R with the property that for every subset S ⊂ M with diamd (S) < ǫ there exists α ∈ A such that S ⊂ Uα . B.2 Fundamental Group and Covering Projections Exercises 1. Show that every Hausdorff locally Euclidean space is regular. 2. Show that every regular second-countable space is normal. 3. Verify Lemma B.1.1. 93 Index chart smooth, 3 cotangent bundle, 59 quotient, 12 smoothly homotopic, 70 submanifold, 24 diffeomorphism, 8 local, 8 distribution, 52 tautological line bundle complex, 42 real, 41 foliation, 52 proper, 52 vector bundle complex, 41 direct sum, 57 dual, 58 exterior product, 62 orientable, 67 quotient, 54 real, 40 section, 42 tensor product, 60 zero section, 43 vector field, 42 flow, 79 immersion, 24 integral, 53 regular, 28 integral curve, 77 Lie bracket, 44 Lie derivative, 83 line bundle, 40 line with two origins, 2 locally Euclidean, 2 manifold orientable, 69 smooth, 3 topological, 2 normal bundle immersion, 55 submanifold, 55 orientation double cover, 70 partition of unity, 65 projective space complex, 6 real, 5 smooth map, 8 smooth structure, 3 product, 7 94 Bibliography [1] R. Borrelli and C. Coleman, Differential Equations: a Modeling Approach, Prentice Hall, 1987. [2] R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. AMS 64 (1958), 87-89. [3] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. Math. 64 (1956), no. 2, 399–405. [4] J. Milnor, Topology from a Differentiable Viewpoint, Princeton University Press, 1997. [5] J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Bull. AMS 65 (1959), 332–334. [6] J. Munkres, Analysis on Manifolds, Westview Press, 1997. [7] J. Munkres, Topology, Prentice Hall, 2000. [8] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387-399. 95